33:
3426:
8778:
8790:
2213:
5123:
3264:
4321:
184:
will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connected
6196:
4645:
punctures. This is most easily visualized by imagining each puncture as being connected by a string to the boundary of the disk; each mapping homomorphism that permutes two of the punctures can then be seen to be a homotopy of the strings, that is, a braiding of these strings.
2010:
4538:
6714:
422:
of a braid consists of that braid which "undoes" whatever the first braid did, which is obtained by flipping a diagram such as the ones above across a vertical line going through its centre. (The first two example braids above are inverses of each other.)
4844:
6343:
3005:
4661:
If a braid is given and one connects the first left-hand item to the first right-hand item using a new string, the second left-hand item to the second right-hand item etc. (without creating any braids in the new strings), one obtains a
3625:
4179:
5840:
1862:
1779:
3827:
249:
3982:
6035:
439:
in fluid flows. The braiding of (2 + 1)-dimensional space-time trajectories formed by motion of physical rods, periodic orbits or "ghost rods", and almost-invariant sets has been used to estimate the
2208:{\displaystyle B_{n}=\left\langle \sigma _{1},\ldots ,\sigma _{n-1}\mid \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1},\sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}\right\rangle ,}
1687:
6459:
239:
205:
3404:
4418:
215:
273:
6586:
4429:
3540:
6594:
6381:
360:
5435:
340:
5703:
so that the function yields a permutation on endpoints—is isomorphic to this wilder group. An interesting fact is that the pure braid group in this group is isomorphic to both the
5118:{\displaystyle \sigma _{i}\left(x_{1},\ldots ,x_{i-1},x_{i},x_{i+1},\ldots ,x_{n}\right)=\left(x_{1},\ldots ,x_{i-1},x_{i+1},x_{i+1}^{-1}x_{i}x_{i+1},x_{i+2},\ldots ,x_{n}\right).}
2810:
350:
938:
6498:
6007:
1552:
2254:
320:
2995:. These transpositions generate the symmetric group, satisfy the braid group relations, and have order 2. This transforms the Artin presentation of the braid group into the
1287:
1278:
1269:
310:
300:
2400:
3871:
2606:
2296:
1948:
1921:
1894:
1517:
1380:
1350:
1320:
900:
6208:
5701:
5655:
3703:
3259:{\displaystyle S_{n}=\left\langle s_{1},\ldots ,s_{n-1}|s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1},s_{i}s_{j}=s_{j}s_{i}{\text{ for }}|i-j|\geq 2,s_{i}^{2}=1\right\rangle .}
5324:
5609:
5547:
2738:
2576:
2544:
1618:
1164:
described two moves on braid diagrams that yield equivalence in the corresponding closed braids. A single-move version of Markov's theorem, was published by in 1997.
5919:
4316:{\displaystyle \sigma _{1}C\mapsto R={\begin{bmatrix}1&1\\0&1\end{bmatrix}}\qquad \sigma _{2}C\mapsto L^{-1}={\begin{bmatrix}1&0\\-1&1\end{bmatrix}}}
6529:
6411:
5970:
5732:
5488:
5363:
4083:
4032:
3734:
3660:
3488:
3453:
3323:-tuples of distinct points of the Euclidean plane. In a pure braid, the beginning and the end of each strand are in the same position. Pure braid groups fit into a
2765:
2705:
2667:
2636:
2511:
2475:
2427:
2375:
2344:
1998:
1444:
1413:
860:
594:
408:
106:
61:
4734:. (In essence, computing the normal form of a braid is the algebraic analogue of "pulling the strands" as illustrated in our second set of images above.) The free
3548:
5514:
5461:
4686:
arises in this fashion from at least one braid; such a braid can be obtained by cutting the link. Since braids can be concretely given as words in the generators
2896:
1598:
1490:
6027:
5943:
5877:
5748:
4052:
4005:
3891:
1971:
1572:
1464:
1113:
1089:
1065:
1043:
1023:
998:
978:
958:
829:
809:
789:
769:
749:
729:
709:
682:
662:
638:
614:
567:
547:
521:
1787:
1704:
4770:
In analogy with the action of the symmetric group by permutations, in various mathematical settings there exists a natural action of the braid group on
3749:
497:. Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition.
6191:{\displaystyle \operatorname {UConf} _{n}(\mathbb {R} ^{2})=\{\{u_{1},...,u_{n}\}:u_{i}\in \mathbb {R} ^{2},u_{i}\neq u_{j}{\text{ for }}i\neq j\}}
1141:, i.e., a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link can be anything from 1 to
3899:
5851:
476:
7700:
5260:
5180:
remains the identity element. It may be checked that the braid group relations are satisfied and this formula indeed defines a group action of
1869:(these relations can be appreciated best by drawing the braid on a piece of paper). It can be shown that all other relations among the braids
1189:
The "braid index" is the least number of strings needed to make a closed braid representation of a link. It is equal to the least number of
6971:
Stremler, Mark A.; Ross, Shane D.; Grover, Piyush; Kumar, Pankaj (2011), "Topological chaos and periodic braiding of almost-cyclic sets",
7978:
4758:
In addition to the word problem, there are several known hard computational problems that could implement braid groups, applications in
1446:. To see this, an arbitrary braid is scanned from left to right for crossings; beginning at the top, whenever a crossing of strands
1632:
7113:
6416:
8155:
7894:
7869:
7200:
7127:
3332:
791:
strings. Since we must require that the strings never pass through each other, it is necessary that we pass to the subspace
8723:
7957:
4533:{\displaystyle v={\begin{bmatrix}0&1\\-1&0\end{bmatrix}},\qquad p={\begin{bmatrix}0&1\\-1&1\end{bmatrix}}.}
4355:
5280:
4752:
4650:
445:
288:
by drawing the first next to the second, identifying the four items in the middle, and connecting corresponding strands:
6538:
1239:
As Magnus says, Hurwitz gave the interpretation of a braid group as the fundamental group of a configuration space (cf.
8642:
8043:
6839:
5549:(i.e., by attaching a trivial strand). This group, however, admits no metrizable topology while remaining continuous.
8826:
7562:
6793:
Cohen, Daniel; Suciu, Alexander (1997). "The Braid
Monodromy of Plane Algebraic Curves and Hyperplane Arrangements".
6468:
5977:
1233:
494:
3500:
189:. Often some strands will have to pass over or under others, and this is crucial: the following two connections are
6709:{\displaystyle \omega _{k,\ell }\omega _{\ell ,m}+\omega _{\ell ,m}\omega _{m,k}+\omega _{m,k}\omega _{k,\ell }=0.}
4708:
1415:
can be written as a composition of a number of these braids and their inverses. In other words, these three braids
223:
On the other hand, two such connections which can be made to look the same by "pulling the strands" are considered
8189:
7747:
5381:
There are many ways to generalize this notion to an infinite number of strands. The simplest way is to take the
451:
Another field of intense investigation involving braid groups and related topological concepts in the context of
6351:
5571:
The second group can be thought of the same as with finite braid groups. Place a strand at each of the points
5272:
1156:
Different braids can give rise to the same link, just as different crossing diagrams can give rise to the same
248:
5388:
5271:, the projective representations of the braid group have a physical meaning for certain quasiparticles in the
8637:
8632:
8508:
7757:
7545:. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore. Vol. 19.
7057:
4735:
500:
To explain how to reduce a braid group in the sense of Artin to a fundamental group, we consider a connected
6903:
Gouillart, Emmanuelle; Thiffeault, Jean-Luc; Finn, Matthew D. (2006), "Topological mixing with ghost rods",
8794:
8209:
7993:
6750:
2780:
525:
17:
185:
with an item of the second set so that a one-to-one correspondence results. Such a connection is called a
8271:
7752:
5880:
3491:
3456:
5259:
described a family of more general "Lawrence representations" depending on several parameters. In 1996,
905:
7119:
4829:
1181:
gives necessary and sufficient conditions under which the closures of two braids are equivalent links.
1146:
6474:
5983:
1522:
8831:
8341:
8336:
8277:
8148:
6730:
5196:
2221:
1416:
4751:
for GAP3 with special support for braid groups. The word problem is also efficiently solved via the
7769:
6338:{\displaystyle H^{*}(B_{n})=H^{*}(K(B_{n},1))=H^{*}(\operatorname {UConf} _{n}(\mathbb {R} ^{2})).}
3631:
Furthermore, the modular group has trivial center, and thus the modular group is isomorphic to the
2307:
1149:
demonstrates that every link can be obtained in this way as the "closure" of a braid. Compare with
1091:
is the
Euclidean plane is the original one of Artin. In some cases it can be shown that the higher
148:
4605:
is in the center, the modular group has trivial center, and the above surjective homomorphism has
2383:
8469:
5241:
4704:
4346:
3836:
2584:
2259:
2001:
1926:
1899:
1872:
1495:
1358:
1328:
1298:
1221:
865:
8683:
8652:
8036:
5660:
5614:
5611:
and the set of all braids—where a braid is defined to be a collection of paths from the points
4671:
3669:
136:
5286:
8816:
8513:
7984:
7882:
7224:
6755:
5574:
5564:
of the infinitely punctured disk—a discrete set of punctures limiting to the boundary of the
5557:
5519:
4339:
3663:
3620:{\displaystyle {\overline {\mathrm {SL} (2,\mathbb {R} )}}\to \mathrm {PSL} (2,\mathbb {R} )}
2717:
2555:
2516:
1603:
1174:
as a braid invariant and then showed that it depended only on the class of the closed braid.
5889:
5279:
and Daan
Krammer independently proved that all braid groups are linear. Their work used the
8821:
8782:
8553:
8141:
7847:
7806:
7778:
7731:
7647:
7602:
7515:
7469:
7403:
7358:
7289:
7137:
7088:
6980:
6942:
6922:
6874:
6854:
6740:
6735:
6725:
6507:
6389:
5948:
5710:
5466:
5366:
5341:
5233:
5229:
5204:
4061:
4010:
3712:
3638:
3466:
3431:
3324:
2928:
elements. This assignment is onto and compatible with composition, and therefore becomes a
2743:
2683:
2645:
2639:
2614:
2484:
2453:
2405:
2353:
2322:
1976:
1422:
1391:
838:
685:
572:
386:
238:
204:
140:
84:
39:
5835:{\displaystyle \{(x_{i})_{i\in \mathbb {N} }\mid x_{i}=x_{j}{\text{ for some }}i\neq j\}.}
1228:
in 1947. Braid groups are also understood by a deeper mathematical interpretation: as the
481:
To put the above informal discussion of braid groups on firm ground, one needs to use the
8:
8590:
8573:
7628:
7384:
Read, N. (2003), "Nonabelian braid statistics versus projective permutation statistics",
5561:
5493:
5440:
4632:
2875:
1577:
1469:
464:
455:
is in the theory and (conjectured) experimental implementation of the proposed particles
441:
411:
7782:
7407:
7362:
6984:
6926:
6858:
135:, where any knot may be represented as the closure of certain braids (a result known as
8611:
8558:
8172:
8168:
8117:
7928:
7810:
7735:
7709:
7677:
7606:
7519:
7493:
7473:
7447:
7419:
7393:
7374:
7348:
7304:
7241:
7092:
7066:
6954:
6912:
6878:
6820:
6802:
6012:
5928:
5862:
5237:
5159:
4747:
if the elements are given in terms of these generators. There is also a package called
4683:
4679:
4667:
4663:
4649:
Via this mapping class group interpretation of braids, each braid may be classified as
4617:
4037:
3990:
3876:
3706:
3410:
2932:
2775:
1956:
1557:
1449:
1157:
1138:
1098:
1074:
1050:
1028:
1008:
983:
963:
943:
814:
794:
774:
754:
734:
714:
694:
667:
647:
623:
599:
552:
532:
506:
486:
160:
7080:
8708:
8657:
8607:
8563:
8523:
8518:
8436:
8029:
7890:
7865:
7814:
7794:
7610:
7558:
7423:
7370:
7196:
7150:
7123:
7025:
7008:
6946:
5883:
5735:
5553:
5200:
2670:
1229:
1161:
617:
490:
460:
214:
7739:
7523:
7484:
Fabel, Paul (2006), "The mapping class group of a disk with infinitely many holes",
7477:
7378:
7096:
6882:
6824:
272:
8743:
8568:
8464:
8199:
8097:
7833:
7786:
7719:
7695:
7590:
7550:
7546:
7503:
7457:
7411:
7382:
Some of
Wilczek-Nayak's proposals subtly violate known physics; see the discussion
7366:
7275:
7233:
7188:
7076:
6998:
6993:
6988:
6958:
6930:
6862:
6812:
6532:
5857:
4811:
4606:
3542:, with these sitting as lattices inside the (topological) universal covering group
3270:
1857:{\displaystyle \sigma _{2}\sigma _{3}\sigma _{2}=\sigma _{3}\sigma _{2}\sigma _{3}}
1774:{\displaystyle \sigma _{1}\sigma _{2}\sigma _{1}=\sigma _{2}\sigma _{1}\sigma _{2}}
1171:
415:
5560:
yields a different group. The first is a very tame group and is isomorphic to the
3822:{\displaystyle a=\sigma _{1}\sigma _{2}\sigma _{1},\quad b=\sigma _{1}\sigma _{2}}
257:
All strands are required to move from left to right; knots like the following are
8703:
8667:
8602:
8548:
8503:
8496:
8386:
8298:
8181:
7953:
7924:
7878:
7859:
7855:
7843:
7802:
7727:
7669:
7643:
7624:
7598:
7511:
7465:
7438:
Fabel, Paul (2005), "Completing Artin's braid group on infinitely many strands",
7285:
7133:
7084:
7052:
6938:
6870:
6501:
5370:
5276:
4113:
4085:
2951:
641:
452:
432:
419:
124:
120:
8133:
8763:
8662:
8624:
8543:
8456:
8331:
8323:
8283:
7838:
7764:
7554:
7280:
7263:
7176:
7154:
6934:
6775:
5208:
4783:
4695:, this is often the preferred method of entering knots into computer programs.
4636:
4117:
3632:
2850:
2708:
1206:
1190:
1178:
1145:, depending on the permutation of strands determined by the link. A theorem of
1124:
1092:
436:
8016:
8004:
7939:
7507:
7461:
6866:
3409:
This sequence splits and therefore pure braid groups are realized as iterated
1600:. Upon reaching the right end, the braid has been written as a product of the
32:
8810:
8698:
8486:
8479:
8474:
8112:
8076:
7798:
7538:
7325:
5704:
5264:
5256:
3495:
2996:
2438:
2347:
1210:
1167:
36:
A regular braid on five strands. Each arrow composes two further elements of
7961:
6886:
5203:
with a braid group action. Such structures play an important role in modern
3425:
444:
of several engineered and naturally occurring fluid systems, via the use of
8713:
8693:
8597:
8580:
8376:
8313:
8071:
7578:
7187:. Lecture Notes in Mathematics. Vol. 372. Springer. pp. 463–487.
7012:
6950:
5739:
5382:
5252:
4759:
4675:
3977:{\displaystyle \sigma _{1}c\sigma _{1}^{-1}=\sigma _{2}c\sigma _{2}^{-1}=c}
2677:
2434:
2378:
1240:
1150:
113:
7723:
7185:
Proceedings of the Second
International Conference on the Theory of Groups
6816:
1243:), an interpretation that was lost from view until it was rediscovered by
8728:
8491:
8396:
8265:
8245:
8235:
8227:
8219:
8164:
7665:
7353:
7109:
6807:
6745:
2919:
2907:
2299:
132:
68:
8102:
7682:
7339:-Dimensional Spinor Braiding Statistics in Paired Quantum Hall States",
5251:. More generally, it was a major open problem whether braid groups were
5240:. It had been a long-standing question whether Burau representation was
8748:
8733:
8688:
8585:
8538:
8533:
8528:
8358:
8255:
8107:
7916:
7790:
7698:; Weiermann, Andreas (2011), "Unprovability results involving braids",
7594:
7398:
7245:
7219:
7192:
7180:
7003:
5922:
3740:
2929:
2768:
2430:
1225:
1202:
359:
144:
7415:
463:
and so their abstract study is currently of fundamental importance in
339:
8753:
8421:
7821:
7498:
7452:
7259:
7071:
6917:
5565:
1244:
1214:
664:
strands operating on the indices of coordinates. That is, an ordered
418:
is the braid consisting of four parallel horizontal strands, and the
156:
7911:
7309:
7237:
4618:
Relationship to the mapping class group and classification of braids
2298:. This presentation leads to generalisations of braid groups called
8738:
8348:
8052:
5852:
Configuration space (mathematics) § Connection to braid groups
5228:
can be represented more concretely by matrices. One classical such
4707:
for the braid relations is efficiently solvable and there exists a
4055:
2478:
501:
482:
477:
Configuration space (mathematics) § Connection to braid groups
7714:
349:
8122:
7541:(1 December 2009). "Configuration Spaces, Braids, and Robotics".
7118:, Annals of Mathematics Studies, vol. 82, Princeton, N.J.:
7029:
4342:; it is well known that these moves generate the modular group.
1137:, i.e., corresponding ends can be connected in pairs, to form a
147:'s canonical presentation of the braid group corresponds to the
8758:
8406:
8366:
2546:
by adding an extra strand that does not cross any of the first
459:. These may well end up forming the basis for error-corrected
456:
8017:
Behind the Math of "Dance Your PhD," Part 1: The Braid Groups.
2912:
By forgetting how the strands twist and cross, every braid on
8647:
8092:
7932:
4151:
2847:
1 + 1 − 1 + 1 + 1 = 3
1682:{\displaystyle \sigma _{1}\sigma _{3}=\sigma _{3}\sigma _{1}}
1068:
319:
7767:(1972), "Les immeubles des groupes de tresses généralisés",
6838:
Boyland, Philip L.; Aref, Hassan; Stremler, Mark A. (2000),
6504:
showed that the integral cohomology of the pure braid group
5338:. By suitably specializing these variables, the braid group
3416:
3317:. This can be seen as the fundamental group of the space of
1695:
while the following two relations are not quite as obvious:
8718:
8021:
6454:{\displaystyle \operatorname {Conf} _{n}(\mathbb {R} ^{2})}
2552:
strands. The increasing union of the braid groups with all
1286:
1277:
1268:
1067:(for any choice of base point – this is well-defined
960:
is the quotient by the symmetric group of the non-excluded
309:
299:
1950:
already follow from these relations and the group axioms.
711:-fold symmetric product is the abstract way of discussing
7693:
7034:
Recueil Mathématique de la Société Mathématique de Moscou
5556:
that can be imposed on the resulting group each of whose
5267:
posited that in analogy to projective representations of
4786:
that involves some "twists". Consider an arbitrary group
112:, is the group whose elements are equivalence classes of
6386:
Similarly, a classifying space for the pure braid group
5945:
up to homotopy. A classifying space for the braid group
6970:
6902:
4493:
4444:
4279:
4210:
4150:; this isomorphism can be given an explicit form. The
3399:{\displaystyle 1\to F_{n-1}\to P_{n}\to P_{n-1}\to 1.}
2908:
Relation with symmetric group and the pure braid group
2898:. This proves that the generators have infinite order.
210: is different from
7877:
7688:
6597:
6541:
6510:
6477:
6419:
6392:
6354:
6211:
6038:
6015:
5986:
5951:
5931:
5892:
5865:
5751:
5713:
5663:
5617:
5577:
5522:
5496:
5469:
5443:
5391:
5344:
5289:
5174:– this ensures that the product of the components of
4847:
4432:
4413:{\displaystyle \langle v,p\,|\,v^{2}=p^{3}=1\rangle }
4358:
4182:
4064:
4040:
4013:
3993:
3902:
3879:
3839:
3752:
3715:
3672:
3641:
3551:
3503:
3469:
3434:
3335:
3008:
2878:
2783:
2746:
2720:
2686:
2648:
2617:
2587:
2558:
2519:
2487:
2456:
2408:
2386:
2356:
2325:
2262:
2224:
2013:
1979:
1959:
1929:
1902:
1875:
1790:
1707:
1635:
1606:
1580:
1560:
1525:
1498:
1472:
1452:
1425:
1394:
1361:
1331:
1301:
1101:
1077:
1053:
1031:
1011:
986:
966:
946:
908:
868:
841:
817:
797:
777:
757:
737:
717:
697:
670:
650:
626:
602:
575:
555:
535:
509:
389:
87:
42:
7941:
Lecture 1.3: Groups in science, art, and mathematics
5376:
383:
The set of all braids on four strands is denoted by
345: composed with
305: composed with
244: is the same as
7030:"Über die freie Äquivalenz der geschlossenen Zöpfe"
6837:
1209:pointed out in 1974) they were already implicit in
940:. This is invariant under the symmetric group, and
7303:Garber, David (2009). "Braid Group Cryptography".
7149:
7050:
6708:
6581:{\displaystyle \omega _{ij}\;\;1\leq i<j\leq n}
6580:
6535:generated by the collection of degree-one classes
6523:
6492:
6453:
6405:
6375:
6337:
6190:
6021:
6001:
5964:
5937:
5913:
5879:is defined as the cohomology of the corresponding
5871:
5834:
5726:
5695:
5649:
5603:
5541:
5508:
5482:
5455:
5429:
5357:
5318:
5117:
4532:
4412:
4315:
4077:
4046:
4026:
3999:
3976:
3885:
3865:
3821:
3728:
3697:
3654:
3619:
3534:
3482:
3447:
3398:
3258:
2890:
2804:
2759:
2732:
2699:
2661:
2630:
2600:
2570:
2538:
2505:
2469:
2421:
2394:
2369:
2338:
2290:
2248:
2207:
1992:
1965:
1942:
1915:
1888:
1856:
1773:
1681:
1612:
1592:
1566:
1546:
1511:
1484:
1458:
1438:
1407:
1374:
1344:
1314:
1107:
1083:
1059:
1037:
1017:
992:
972:
952:
932:
894:
854:
823:
803:
783:
763:
743:
723:
703:
676:
656:
632:
608:
588:
561:
541:
515:
402:
100:
55:
8163:
688:as any other that is a re-ordered version of it.
8808:
7629:"The cohomology ring of the colored braid group"
4678:states that the converse is true as well: every
835:points. That is, we remove all the subspaces of
131:). Example applications of braid groups include
27:Group whose operation is a composition of braids
7581:(1970). "Cohomology of the braid group mod 2".
5244:, but the answer turned out to be negative for
5236:, where the matrix entries are single variable
3893:, one may verify from the braid relations that
7701:Proceedings of the London Mathematical Society
3535:{\displaystyle \mathrm {PSL} (2,\mathbb {Z} )}
410:. The above composition of braids is indeed a
8149:
8037:
7664:
4338:are the standard left and right moves on the
1554:is written down, depending on whether strand
355: yields
315: yields
7853:
7824:; Neuwirth, Lee (1962), "The braid groups",
7486:Journal of Knot Theory and Its Ramifications
7440:Journal of Knot Theory and Its Ramifications
7323:
6185:
6114:
6076:
6073:
5826:
5752:
4656:
4407:
4359:
2000:can be abstractly defined via the following
7820:
7745:
7623:
7258:
7055:(1997), "Markov's theorem in 3-manifolds",
1255:
1201:Braid groups were introduced explicitly by
8156:
8142:
8044:
8030:
7214:
7212:
6792:
6556:
6555:
6376:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
5385:of braid groups, where the attaching maps
2306:, play an important role in the theory of
1220:Braid groups may be described by explicit
431:Braid theory has recently been applied to
7837:
7713:
7681:
7497:
7451:
7397:
7352:
7308:
7279:
7070:
7002:
6992:
6916:
6840:"Topological fluid mechanics of stirring"
6806:
6480:
6438:
6369:
6356:
6316:
6135:
6057:
6029:distinct unordered points in the plane:
5989:
5780:
4377:
4371:
3833:From the braid relations it follows that
3610:
3573:
3525:
2798:
2388:
7583:Functional Analysis and Its Applications
5552:Paul Fabel has shown that there are two
5430:{\displaystyle f\colon B_{n}\to B_{n+1}}
4698:
3424:
1003:With this definition, then, we can call
178:; the generalization to other values of
31:
7763:
7209:
7115:Braids, links, and mapping class groups
4567:yields a surjective group homomorphism
980:-tuples. Under the dimension condition
811:of the symmetric product, of orbits of
14:
8809:
7991:
7537:
7302:
7175:
7143:
7108:
7024:
2767:contains a subgroup isomorphic to the
2256:and in the second group of relations,
2218:where in the first group of relations
8137:
8025:
7483:
7437:
7218:
6773:
6348:The calculations for coefficients in
5365:may be realized as a subgroup of the
4631:can be shown to be isomorphic to the
2805:{\displaystyle B_{n}\to \mathbb {Z} }
2313:
1260:Consider the following three braids:
152:
8789:
8002:
7937:
7577:
7383:
7157:. MathWorld – A Wolfram Web Resource
4651:periodic, reducible or pseudo-Anosov
2302:. The cubic relations, known as the
8006:Representations of the Braid Groups
7952:
2437:– in particular, it is an infinite
1250:
470:
24:
7980:Cryptography and Braid Groups page
7976:
7657:
5214:
5149:exchange places and, in addition,
4599:, a consequence of the facts that
3873:. Denoting this latter product as
3705:and equivalently, to the group of
3596:
3593:
3590:
3559:
3556:
3511:
3508:
3505:
2593:
933:{\displaystyle 1\leq i<j\leq n}
771:-tuple, independently tracing out
25:
8843:
7904:
7689:Menasco & Thistlethwaite 2005
6795:Commentarii Mathematici Helvetici
5377:Infinitely generated braid groups
640:by the permutation action of the
278: is not a braid
128:
8788:
8777:
8776:
6493:{\displaystyle \mathbb {R} ^{2}}
6002:{\displaystyle \mathbb {R} ^{2}}
2872:is the identity if and only if
1547:{\displaystyle \sigma _{i}^{-1}}
1285:
1276:
1267:
1118:
358:
348:
338:
318:
308:
298:
271:
247:
237:
213:
203:
7958:"Exploration of B5 Java applet"
7617:
7571:
7531:
7429:
7386:Journal of Mathematical Physics
7317:
7296:
7252:
7169:
5281:Lawrence–Krammer representation
4753:Lawrence–Krammer representation
4481:
4241:
3792:
3739:Here is a construction of this
2902:
2249:{\displaystyle 1\leq i\leq n-2}
1133:is the plane, the braid can be
446:Nielsen–Thurston classification
435:, specifically to the field of
426:
166:
8643:Dowker–Thistlethwaite notation
7102:
7044:
7018:
6994:10.1103/PhysRevLett.106.114101
6964:
6896:
6831:
6786:
6767:
6448:
6433:
6329:
6326:
6311:
6295:
6279:
6276:
6257:
6251:
6235:
6222:
6067:
6052:
5908:
5896:
5769:
5755:
5690:
5664:
5644:
5618:
5598:
5578:
5408:
5305:
5293:
5273:fractional quantum hall effect
4738:can carry out computations in
4373:
4255:
4196:
3689:
3676:
3614:
3600:
3586:
3577:
3563:
3529:
3515:
3390:
3371:
3358:
3339:
3214:
3200:
3063:
2950:from the braid group onto the
2849:. This map corresponds to the
2794:
2500:
2488:
2278:
2264:
1184:
368:The composition of the braids
127:is composition of braids (see
13:
1:
7933:Algebraic Cryptography Center
7927:computation library from the
7925:CRAG: CRyptography and Groups
7081:10.1016/S0166-8641(96)00151-4
7058:Topology and Its Applications
6761:
6383:can be found in Fuks (1970).
5845:
4776:-tuples of objects or on the
2822:. So for instance, the braid
2611:All non-identity elements of
1953:Generalising this example to
1193:in any projection of a knot.
1071:isomorphism). The case where
751:, considered as an unordered
523:of dimension at least 2. The
8051:
7371:10.1016/0550-3213(96)00430-0
7222:(1947). "Theory of Braids".
6751:Non-commutative cryptography
5707:of finite pure braid groups
5219:Elements of the braid group
3581:
2395:{\displaystyle \mathbb {Z} }
7:
7753:Encyclopedia of Mathematics
7746:Chernavskii, A.V. (2001) ,
6719:
6588:, subject to the relations
5326:depending on the variables
4736:GAP computer algebra system
4720:in terms of the generators
3866:{\displaystyle a^{2}=b^{3}}
3492:universal central extension
3457:universal central extension
2601:{\displaystyle B_{\infty }}
2291:{\displaystyle |i-j|\geq 2}
1943:{\displaystyle \sigma _{3}}
1916:{\displaystyle \sigma _{2}}
1889:{\displaystyle \sigma _{1}}
1574:moves under or over strand
1512:{\displaystyle \sigma _{i}}
1375:{\displaystyle \sigma _{3}}
1345:{\displaystyle \sigma _{2}}
1315:{\displaystyle \sigma _{1}}
895:{\displaystyle x_{i}=x_{j}}
489:, defining braid groups as
10:
8848:
7839:10.7146/math.scand.a-10518
7555:10.1142/9789814291415_0004
7281:10.7146/math.scand.a-10518
7120:Princeton University Press
6935:10.1103/PhysRevE.73.036311
6847:Journal of Fluid Mechanics
5849:
4838:in the following fashion:
4765:
2954:. The image of the braid σ
2853:of the braid group. Since
2676:There is a left-invariant
1247:and Lee Neuwirth in 1962.
1196:
1122:
474:
8772:
8676:
8633:Alexander–Briggs notation
8620:
8455:
8357:
8322:
8180:
8085:
8059:
7508:10.1142/S0218216506004324
7462:10.1142/S0218216505004196
7335:Quasihole States Realize
7036:(in German and Russian),
6867:10.1017/S0022112099007107
6731:Braided monoidal category
5696:{\displaystyle (0,1/n,1)}
5650:{\displaystyle (0,1/n,0)}
5197:braided monoidal category
4657:Connection to knot theory
4349:for the modular group is
3698:{\displaystyle Z(B_{3}),}
1047:the fundamental group of
171:In this introduction let
8827:Low-dimensional topology
7826:Mathematica Scandinavica
7770:Inventiones Mathematicae
7268:Mathematica Scandinavica
7262:; Neuwirth, Lee (1962).
7181:"Braid groups: A survey"
5319:{\displaystyle n(n-1)/2}
5195:. As another example, a
4345:Alternately, one common
2999:of the symmetric group:
1256:Generators and relations
8724:List of knots and links
8272:Kinoshita–Terasaka knot
7887:Handbook of Knot Theory
6973:Physical Review Letters
6531:is the quotient of the
5925:uniquely determined by
5604:{\displaystyle (0,1/n)}
5542:{\displaystyle B_{n+1}}
4804:-tuples of elements of
2733:{\displaystyle n\geq 3}
2571:{\displaystyle n\geq 1}
2539:{\displaystyle B_{n+1}}
1620:'s and their inverses.
1613:{\displaystyle \sigma }
1170:originally defined his
153:§ Basic properties
7992:Dalvit, Ester (2015).
7883:Thistlethwaite, Morwen
6710:
6582:
6525:
6494:
6455:
6407:
6377:
6339:
6192:
6023:
6009:, that is, the set of
6003:
5966:
5939:
5915:
5914:{\displaystyle K(G,1)}
5873:
5836:
5728:
5697:
5651:
5605:
5543:
5510:
5484:
5457:
5431:
5359:
5320:
5119:
4534:
4414:
4317:
4079:
4048:
4028:
4001:
3978:
3887:
3867:
3823:
3730:
3699:
3656:
3621:
3536:
3484:
3460:
3449:
3400:
3260:
2892:
2806:
2761:
2734:
2701:
2663:
2632:
2602:
2572:
2540:
2507:
2471:
2423:
2396:
2371:
2340:
2292:
2250:
2209:
1994:
1967:
1944:
1917:
1890:
1858:
1775:
1683:
1614:
1594:
1568:
1548:
1513:
1486:
1460:
1440:
1409:
1376:
1346:
1316:
1205:in 1925, although (as
1109:
1085:
1061:
1039:
1019:
994:
974:
954:
934:
896:
862:defined by conditions
856:
825:
805:
785:
765:
745:
725:
705:
684:-tuple is in the same
678:
658:
634:
610:
590:
569:means the quotient of
563:
543:
517:
404:
284:Any two braids can be
102:
64:
57:
8514:Finite type invariant
7948:. Clemson University.
7225:Annals of Mathematics
7051:Lambropoulou, Sofia;
6817:10.1007/s000140050017
6756:Spherical braid group
6711:
6583:
6526:
6524:{\displaystyle P_{n}}
6495:
6456:
6408:
6406:{\displaystyle P_{n}}
6378:
6340:
6193:
6024:
6004:
5967:
5965:{\displaystyle B_{n}}
5940:
5916:
5874:
5858:cohomology of a group
5837:
5729:
5727:{\displaystyle P_{n}}
5698:
5652:
5606:
5544:
5511:
5485:
5483:{\displaystyle B_{n}}
5458:
5432:
5360:
5358:{\displaystyle B_{n}}
5321:
5120:
4810:whose product is the
4762:have been suggested.
4699:Computational aspects
4535:
4415:
4318:
4116:and one may take the
4080:
4078:{\displaystyle B_{3}}
4049:
4029:
4027:{\displaystyle B_{3}}
4002:
3979:
3888:
3868:
3824:
3731:
3729:{\displaystyle B_{3}}
3700:
3657:
3655:{\displaystyle B_{3}}
3622:
3537:
3485:
3483:{\displaystyle B_{3}}
3459:of the modular group.
3450:
3448:{\displaystyle B_{3}}
3428:
3421:and the modular group
3401:
3261:
2969:is the transposition
2918:strands determines a
2893:
2807:
2762:
2760:{\displaystyle B_{n}}
2735:
2702:
2700:{\displaystyle B_{n}}
2664:
2662:{\displaystyle B_{n}}
2633:
2631:{\displaystyle B_{n}}
2603:
2573:
2541:
2508:
2506:{\displaystyle (n+1)}
2472:
2470:{\displaystyle B_{n}}
2429:is isomorphic to the
2424:
2422:{\displaystyle B_{3}}
2397:
2372:
2370:{\displaystyle B_{2}}
2341:
2339:{\displaystyle B_{1}}
2308:Yang–Baxter equations
2293:
2251:
2210:
1995:
1993:{\displaystyle B_{n}}
1968:
1945:
1918:
1891:
1859:
1776:
1684:
1615:
1595:
1569:
1549:
1514:
1487:
1461:
1441:
1439:{\displaystyle B_{4}}
1410:
1408:{\displaystyle B_{4}}
1377:
1347:
1317:
1110:
1086:
1062:
1040:
1020:
995:
975:
955:
935:
897:
857:
855:{\displaystyle X^{n}}
826:
806:
786:
766:
746:
726:
706:
679:
659:
635:
611:
591:
589:{\displaystyle X^{n}}
564:
544:
518:
405:
403:{\displaystyle B_{4}}
108:), also known as the
103:
101:{\displaystyle B_{n}}
58:
56:{\displaystyle B_{5}}
35:
8015:expanded further in
7672:(26 February 2005),
7549:. pp. 263–304.
6741:Braided Hopf algebra
6736:Braided vector space
6595:
6539:
6508:
6475:
6417:
6390:
6352:
6209:
6036:
6013:
5984:
5949:
5929:
5890:
5863:
5814: for some
5749:
5711:
5661:
5615:
5575:
5520:
5494:
5467:
5441:
5389:
5367:general linear group
5342:
5287:
5234:Burau representation
5207:and lead to quantum
5205:mathematical physics
4845:
4430:
4356:
4180:
4062:
4038:
4011:
4007:is in the center of
3991:
3900:
3877:
3837:
3750:
3713:
3670:
3639:
3549:
3501:
3467:
3432:
3411:semi-direct products
3333:
3325:short exact sequence
3300:pure braid group on
3273:of the homomorphism
3006:
2997:Coxeter presentation
2876:
2781:
2744:
2718:
2684:
2646:
2615:
2585:
2580:infinite braid group
2556:
2517:
2513:-strand braid group
2485:
2454:
2450:-strand braid group
2406:
2384:
2354:
2323:
2260:
2222:
2011:
1977:
1957:
1927:
1900:
1873:
1788:
1705:
1633:
1604:
1578:
1558:
1523:
1496:
1470:
1450:
1423:
1392:
1359:
1329:
1299:
1234:configuration spaces
1099:
1075:
1051:
1029:
1009:
984:
964:
944:
906:
866:
839:
815:
795:
775:
755:
735:
715:
695:
668:
648:
624:
600:
573:
553:
533:
507:
387:
149:Yang–Baxter equation
141:mathematical physics
85:
40:
8684:Alexander's theorem
7946:Visual Group Theory
7854:Kassel, Christian;
7783:1972InMat..17..273D
7724:10.1112/plms/pdq016
7694:Carlucci, Lorenzo;
7408:2003JMP....44..558R
7363:1996NuPhB.479..529N
6985:2011PhRvL.106k4101S
6927:2006PhRvE..73c6311G
6859:2000JFM...403..277B
6469:configuration space
5978:configuration space
5562:mapping class group
5509:{\displaystyle n-1}
5456:{\displaystyle n-1}
5238:Laurent polynomials
5042:
4672:Alexander's theorem
4633:mapping class group
3967:
3933:
3707:inner automorphisms
3289:is the subgroup of
3241:
2891:{\displaystyle k=0}
1973:strands, the group
1593:{\displaystyle i+1}
1543:
1485:{\displaystyle i+1}
1290:
1281:
1272:
1005:the braid group of
1000:will be connected.
495:configuration space
465:quantum information
442:topological entropy
261:considered braids:
137:Alexander's theorem
129:§ Introduction
8118:3D braided fabrics
7995:Braids – the movie
7929:Stevens University
7791:10.1007/BF01406236
7595:10.1007/BF01094491
7264:"The braid groups"
7193:10.1007/BFb0065203
7151:Weisstein, Eric W.
6706:
6578:
6521:
6490:
6451:
6403:
6373:
6335:
6188:
6019:
5999:
5962:
5935:
5911:
5869:
5832:
5724:
5693:
5647:
5601:
5539:
5506:
5480:
5453:
5427:
5355:
5316:
5160:inner automorphism
5158:is twisted by the
5128:Thus the elements
5115:
5019:
4798:be the set of all
4666:, and sometimes a
4530:
4521:
4472:
4410:
4313:
4307:
4235:
4075:
4044:
4024:
3997:
3974:
3950:
3916:
3883:
3863:
3819:
3726:
3695:
3652:
3617:
3532:
3480:
3461:
3445:
3417:Relation between B
3396:
3256:
3227:
2933:group homomorphism
2888:
2802:
2771:on two generators.
2757:
2730:
2697:
2659:
2628:
2598:
2568:
2536:
2503:
2467:
2419:
2392:
2367:
2336:
2314:Further properties
2288:
2246:
2205:
1990:
1963:
1940:
1913:
1886:
1854:
1771:
1679:
1610:
1590:
1564:
1544:
1526:
1509:
1482:
1456:
1436:
1405:
1372:
1342:
1312:
1284:
1275:
1266:
1224:, as was shown by
1105:
1081:
1057:
1035:
1015:
990:
970:
950:
930:
892:
852:
821:
801:
781:
761:
741:
721:
701:
674:
654:
630:
606:
586:
559:
539:
513:
491:fundamental groups
487:algebraic topology
400:
161:algebraic geometry
98:
65:
53:
8804:
8803:
8658:Reidemeister move
8524:Khovanov homology
8519:Hyperbolic volume
8131:
8130:
8003:Scherich, Nancy.
7896:978-0-444-51452-3
7871:978-0-387-33841-5
7696:Dehornoy, Patrick
7625:Arnol'd, Vladimir
7416:10.1063/1.1530369
7341:Nuclear Physics B
7202:978-3-540-06845-7
7129:978-0-691-08149-6
6905:Physical Review E
6780:Wolfram Mathworld
6774:Weisstein, Eric.
6202:So by definition
6174:
6022:{\displaystyle n}
5938:{\displaystyle G}
5884:classifying space
5881:Eilenberg–MacLane
5872:{\displaystyle G}
5815:
5736:fundamental group
5201:monoidal category
5162:corresponding to
4340:Stern–Brocot tree
4047:{\displaystyle C}
4000:{\displaystyle c}
3886:{\displaystyle c}
3584:
3197:
2439:non-abelian group
1966:{\displaystyle n}
1623:It is clear that
1567:{\displaystyle i}
1459:{\displaystyle i}
1386:
1385:
1230:fundamental group
1162:Andrey Markov Jr.
1108:{\displaystyle Y}
1084:{\displaystyle X}
1060:{\displaystyle Y}
1038:{\displaystyle n}
1018:{\displaystyle X}
993:{\displaystyle Y}
973:{\displaystyle n}
953:{\displaystyle Y}
824:{\displaystyle n}
804:{\displaystyle Y}
784:{\displaystyle n}
764:{\displaystyle n}
744:{\displaystyle X}
724:{\displaystyle n}
704:{\displaystyle n}
677:{\displaystyle n}
657:{\displaystyle n}
633:{\displaystyle X}
618:Cartesian product
609:{\displaystyle n}
562:{\displaystyle X}
542:{\displaystyle n}
526:symmetric product
516:{\displaystyle X}
461:quantum computing
366:
365:
328:Another example:
326:
325:
282:
281:
255:
254:
221:
220:
110:Artin braid group
16:(Redirected from
8839:
8832:Diagram algebras
8792:
8791:
8780:
8779:
8744:Tait conjectures
8447:
8446:
8432:
8431:
8417:
8416:
8309:
8308:
8294:
8293:
8278:(−2,3,7) pretzel
8158:
8151:
8144:
8135:
8134:
8098:Braiding machine
8046:
8039:
8032:
8023:
8022:
8014:
7999:
7988:
7987:on 3 August 2009
7983:, archived from
7977:Lipmaa, Helger,
7973:
7971:
7969:
7960:. Archived from
7954:Bigelow, Stephen
7949:
7921:
7899:
7879:Menasco, William
7874:
7856:Turaev, Vladimir
7850:
7841:
7817:
7760:
7742:
7717:
7686:
7685:
7674:Braids: A Survey
7670:Brendle, Tara E.
7652:
7651:
7633:
7621:
7615:
7614:
7575:
7569:
7568:
7547:World Scientific
7535:
7529:
7526:
7501:
7480:
7455:
7433:
7427:
7426:
7401:
7381:
7356:
7354:cond-mat/9605145
7338:
7334:
7321:
7315:
7314:
7312:
7300:
7294:
7293:
7283:
7256:
7250:
7249:
7216:
7207:
7206:
7173:
7167:
7166:
7164:
7162:
7147:
7141:
7140:
7106:
7100:
7099:
7074:
7053:Rourke, Colin P.
7048:
7042:
7041:
7022:
7016:
7015:
7006:
6996:
6968:
6962:
6961:
6920:
6900:
6894:
6893:
6891:
6885:, archived from
6844:
6835:
6829:
6828:
6810:
6808:alg-geom/9608001
6790:
6784:
6783:
6771:
6726:Artin–Tits group
6715:
6713:
6712:
6707:
6699:
6698:
6683:
6682:
6664:
6663:
6648:
6647:
6629:
6628:
6613:
6612:
6587:
6585:
6584:
6579:
6554:
6553:
6533:exterior algebra
6530:
6528:
6527:
6522:
6520:
6519:
6499:
6497:
6496:
6491:
6489:
6488:
6483:
6460:
6458:
6457:
6452:
6447:
6446:
6441:
6429:
6428:
6412:
6410:
6409:
6404:
6402:
6401:
6382:
6380:
6379:
6374:
6372:
6364:
6359:
6344:
6342:
6341:
6336:
6325:
6324:
6319:
6307:
6306:
6294:
6293:
6269:
6268:
6250:
6249:
6234:
6233:
6221:
6220:
6197:
6195:
6194:
6189:
6175:
6172:
6170:
6169:
6157:
6156:
6144:
6143:
6138:
6129:
6128:
6113:
6112:
6088:
6087:
6066:
6065:
6060:
6048:
6047:
6028:
6026:
6025:
6020:
6008:
6006:
6005:
6000:
5998:
5997:
5992:
5971:
5969:
5968:
5963:
5961:
5960:
5944:
5942:
5941:
5936:
5920:
5918:
5917:
5912:
5878:
5876:
5875:
5870:
5841:
5839:
5838:
5833:
5816:
5813:
5811:
5810:
5798:
5797:
5785:
5784:
5783:
5767:
5766:
5733:
5731:
5730:
5725:
5723:
5722:
5702:
5700:
5699:
5694:
5680:
5656:
5654:
5653:
5648:
5634:
5610:
5608:
5607:
5602:
5594:
5548:
5546:
5545:
5540:
5538:
5537:
5515:
5513:
5512:
5507:
5489:
5487:
5486:
5481:
5479:
5478:
5462:
5460:
5459:
5454:
5436:
5434:
5433:
5428:
5426:
5425:
5407:
5406:
5364:
5362:
5361:
5356:
5354:
5353:
5337:
5331:
5325:
5323:
5322:
5317:
5312:
5270:
5250:
5227:
5194:
5188:
5179:
5173:
5157:
5148:
5136:
5124:
5122:
5121:
5116:
5111:
5107:
5106:
5105:
5087:
5086:
5068:
5067:
5052:
5051:
5041:
5033:
5015:
5014:
4996:
4995:
4971:
4970:
4953:
4949:
4948:
4947:
4929:
4928:
4910:
4909:
4897:
4896:
4872:
4871:
4857:
4856:
4837:
4828:
4819:
4812:identity element
4809:
4803:
4797:
4791:
4781:
4775:
4746:
4733:
4719:
4711:for elements of
4694:
4644:
4630:
4622:The braid group
4613:
4604:
4598:
4592:
4580:
4566:
4560:
4554:
4548:
4539:
4537:
4536:
4531:
4526:
4525:
4477:
4476:
4419:
4417:
4416:
4411:
4400:
4399:
4387:
4386:
4376:
4337:
4331:
4322:
4320:
4319:
4314:
4312:
4311:
4270:
4269:
4251:
4250:
4240:
4239:
4192:
4191:
4172:
4162:
4149:
4131:
4111:
4093:
4084:
4082:
4081:
4076:
4074:
4073:
4053:
4051:
4050:
4045:
4033:
4031:
4030:
4025:
4023:
4022:
4006:
4004:
4003:
3998:
3983:
3981:
3980:
3975:
3966:
3958:
3946:
3945:
3932:
3924:
3912:
3911:
3892:
3890:
3889:
3884:
3872:
3870:
3869:
3864:
3862:
3861:
3849:
3848:
3828:
3826:
3825:
3820:
3818:
3817:
3808:
3807:
3788:
3787:
3778:
3777:
3768:
3767:
3735:
3733:
3732:
3727:
3725:
3724:
3704:
3702:
3701:
3696:
3688:
3687:
3661:
3659:
3658:
3653:
3651:
3650:
3626:
3624:
3623:
3618:
3613:
3599:
3585:
3580:
3576:
3562:
3553:
3541:
3539:
3538:
3533:
3528:
3514:
3489:
3487:
3486:
3481:
3479:
3478:
3463:The braid group
3454:
3452:
3451:
3446:
3444:
3443:
3413:of free groups.
3405:
3403:
3402:
3397:
3389:
3388:
3370:
3369:
3357:
3356:
3322:
3316:
3305:
3297:
3288:
3265:
3263:
3262:
3257:
3252:
3248:
3240:
3235:
3217:
3203:
3198:
3195:
3193:
3192:
3183:
3182:
3170:
3169:
3160:
3159:
3147:
3146:
3131:
3130:
3121:
3120:
3102:
3101:
3092:
3091:
3076:
3075:
3066:
3061:
3060:
3036:
3035:
3018:
3017:
2994:
2968:
2949:
2927:
2917:
2897:
2895:
2894:
2889:
2871:
2862:
2848:
2844:
2821:
2811:
2809:
2808:
2803:
2801:
2793:
2792:
2766:
2764:
2763:
2758:
2756:
2755:
2739:
2737:
2736:
2731:
2706:
2704:
2703:
2698:
2696:
2695:
2668:
2666:
2665:
2660:
2658:
2657:
2637:
2635:
2634:
2629:
2627:
2626:
2607:
2605:
2604:
2599:
2597:
2596:
2577:
2575:
2574:
2569:
2551:
2545:
2543:
2542:
2537:
2535:
2534:
2512:
2510:
2509:
2504:
2476:
2474:
2473:
2468:
2466:
2465:
2449:
2428:
2426:
2425:
2420:
2418:
2417:
2401:
2399:
2398:
2393:
2391:
2377:is the infinite
2376:
2374:
2373:
2368:
2366:
2365:
2345:
2343:
2342:
2337:
2335:
2334:
2319:The braid group
2297:
2295:
2294:
2289:
2281:
2267:
2255:
2253:
2252:
2247:
2214:
2212:
2211:
2206:
2201:
2197:
2196:
2195:
2186:
2185:
2173:
2172:
2163:
2162:
2150:
2149:
2134:
2133:
2124:
2123:
2105:
2104:
2095:
2094:
2079:
2078:
2066:
2065:
2041:
2040:
2023:
2022:
1999:
1997:
1996:
1991:
1989:
1988:
1972:
1970:
1969:
1964:
1949:
1947:
1946:
1941:
1939:
1938:
1922:
1920:
1919:
1914:
1912:
1911:
1895:
1893:
1892:
1887:
1885:
1884:
1863:
1861:
1860:
1855:
1853:
1852:
1843:
1842:
1833:
1832:
1820:
1819:
1810:
1809:
1800:
1799:
1780:
1778:
1777:
1772:
1770:
1769:
1760:
1759:
1750:
1749:
1737:
1736:
1727:
1726:
1717:
1716:
1688:
1686:
1685:
1680:
1678:
1677:
1668:
1667:
1655:
1654:
1645:
1644:
1619:
1617:
1616:
1611:
1599:
1597:
1596:
1591:
1573:
1571:
1570:
1565:
1553:
1551:
1550:
1545:
1542:
1534:
1518:
1516:
1515:
1510:
1508:
1507:
1492:is encountered,
1491:
1489:
1488:
1483:
1465:
1463:
1462:
1457:
1445:
1443:
1442:
1437:
1435:
1434:
1414:
1412:
1411:
1406:
1404:
1403:
1381:
1379:
1378:
1373:
1371:
1370:
1351:
1349:
1348:
1343:
1341:
1340:
1321:
1319:
1318:
1313:
1311:
1310:
1289:
1280:
1271:
1263:
1262:
1251:Basic properties
1114:
1112:
1111:
1106:
1090:
1088:
1087:
1082:
1066:
1064:
1063:
1058:
1044:
1042:
1041:
1036:
1024:
1022:
1021:
1016:
999:
997:
996:
991:
979:
977:
976:
971:
959:
957:
956:
951:
939:
937:
936:
931:
901:
899:
898:
893:
891:
890:
878:
877:
861:
859:
858:
853:
851:
850:
830:
828:
827:
822:
810:
808:
807:
802:
790:
788:
787:
782:
770:
768:
767:
762:
750:
748:
747:
742:
730:
728:
727:
722:
710:
708:
707:
702:
683:
681:
680:
675:
663:
661:
660:
655:
639:
637:
636:
631:
615:
613:
612:
607:
595:
593:
592:
587:
585:
584:
568:
566:
565:
560:
548:
546:
545:
540:
522:
520:
519:
514:
471:Formal treatment
416:identity element
409:
407:
406:
401:
399:
398:
379:
375:
371:
362:
352:
342:
335:
334:
322:
312:
302:
295:
294:
275:
268:
267:
251:
241:
234:
233:
217:
207:
200:
199:
183:
177:
116:
107:
105:
104:
99:
97:
96:
78:
62:
60:
59:
54:
52:
51:
21:
8847:
8846:
8842:
8841:
8840:
8838:
8837:
8836:
8807:
8806:
8805:
8800:
8768:
8672:
8638:Conway notation
8622:
8616:
8603:Tricolorability
8451:
8445:
8442:
8441:
8440:
8430:
8427:
8426:
8425:
8415:
8412:
8411:
8410:
8402:
8392:
8382:
8372:
8353:
8332:Composite knots
8318:
8307:
8304:
8303:
8302:
8299:Borromean rings
8292:
8289:
8288:
8287:
8261:
8251:
8241:
8231:
8223:
8215:
8205:
8195:
8176:
8162:
8132:
8127:
8081:
8055:
8050:
7967:
7965:
7910:
7907:
7902:
7897:
7885:, eds. (2005),
7872:
7765:Deligne, Pierre
7683:math.GT/0409205
7660:
7658:Further reading
7655:
7631:
7622:
7618:
7579:Fuks, Dmitry B.
7576:
7572:
7565:
7536:
7532:
7434:
7430:
7336:
7329:
7324:Nayak, Chetan;
7322:
7318:
7301:
7297:
7257:
7253:
7238:10.2307/1969218
7217:
7210:
7203:
7177:Magnus, Wilhelm
7174:
7170:
7160:
7158:
7153:(August 2014).
7148:
7144:
7130:
7110:Birman, Joan S.
7107:
7103:
7065:(1–2): 95–122,
7049:
7045:
7023:
7019:
6969:
6965:
6901:
6897:
6892:on 26 July 2011
6889:
6842:
6836:
6832:
6791:
6787:
6772:
6768:
6764:
6722:
6688:
6684:
6672:
6668:
6653:
6649:
6637:
6633:
6618:
6614:
6602:
6598:
6596:
6593:
6592:
6546:
6542:
6540:
6537:
6536:
6515:
6511:
6509:
6506:
6505:
6502:Vladimir Arnold
6484:
6479:
6478:
6476:
6473:
6472:
6464:
6442:
6437:
6436:
6424:
6420:
6418:
6415:
6414:
6397:
6393:
6391:
6388:
6387:
6368:
6360:
6355:
6353:
6350:
6349:
6320:
6315:
6314:
6302:
6298:
6289:
6285:
6264:
6260:
6245:
6241:
6229:
6225:
6216:
6212:
6210:
6207:
6206:
6173: for
6171:
6165:
6161:
6152:
6148:
6139:
6134:
6133:
6124:
6120:
6108:
6104:
6083:
6079:
6061:
6056:
6055:
6043:
6039:
6037:
6034:
6033:
6014:
6011:
6010:
5993:
5988:
5987:
5985:
5982:
5981:
5975:
5956:
5952:
5950:
5947:
5946:
5930:
5927:
5926:
5891:
5888:
5887:
5864:
5861:
5860:
5854:
5848:
5812:
5806:
5802:
5793:
5789:
5779:
5772:
5768:
5762:
5758:
5750:
5747:
5746:
5718:
5714:
5712:
5709:
5708:
5676:
5662:
5659:
5658:
5630:
5616:
5613:
5612:
5590:
5576:
5573:
5572:
5527:
5523:
5521:
5518:
5517:
5495:
5492:
5491:
5474:
5470:
5468:
5465:
5464:
5442:
5439:
5438:
5415:
5411:
5402:
5398:
5390:
5387:
5386:
5379:
5371:complex numbers
5349:
5345:
5343:
5340:
5339:
5333:
5327:
5308:
5288:
5285:
5284:
5277:Stephen Bigelow
5268:
5245:
5225:
5220:
5217:
5215:Representations
5209:knot invariants
5190:
5186:
5181:
5175:
5172:
5163:
5155:
5150:
5147:
5138:
5134:
5129:
5101:
5097:
5076:
5072:
5057:
5053:
5047:
5043:
5034:
5023:
5004:
5000:
4985:
4981:
4966:
4962:
4961:
4957:
4943:
4939:
4918:
4914:
4905:
4901:
4886:
4882:
4867:
4863:
4862:
4858:
4852:
4848:
4846:
4843:
4842:
4833:
4826:
4821:
4815:
4805:
4799:
4793:
4787:
4777:
4771:
4768:
4744:
4739:
4732:
4725:
4721:
4717:
4712:
4701:
4693:
4687:
4659:
4640:
4628:
4623:
4620:
4609:
4600:
4594:
4591:
4585:
4574:
4568:
4562:
4556:
4550:
4544:
4520:
4519:
4514:
4505:
4504:
4499:
4489:
4488:
4471:
4470:
4465:
4456:
4455:
4450:
4440:
4439:
4431:
4428:
4427:
4395:
4391:
4382:
4378:
4372:
4357:
4354:
4353:
4333:
4327:
4306:
4305:
4300:
4291:
4290:
4285:
4275:
4274:
4262:
4258:
4246:
4242:
4234:
4233:
4228:
4222:
4221:
4216:
4206:
4205:
4187:
4183:
4181:
4178:
4177:
4168:
4164:
4158:
4154:
4139:
4133:
4126:
4120:
4114:normal subgroup
4109:
4095:
4089:
4069:
4065:
4063:
4060:
4059:
4039:
4036:
4035:
4018:
4014:
4012:
4009:
4008:
3992:
3989:
3988:
3959:
3954:
3941:
3937:
3925:
3920:
3907:
3903:
3901:
3898:
3897:
3878:
3875:
3874:
3857:
3853:
3844:
3840:
3838:
3835:
3834:
3813:
3809:
3803:
3799:
3783:
3779:
3773:
3769:
3763:
3759:
3751:
3748:
3747:
3720:
3716:
3714:
3711:
3710:
3683:
3679:
3671:
3668:
3667:
3646:
3642:
3640:
3637:
3636:
3609:
3589:
3572:
3555:
3554:
3552:
3550:
3547:
3546:
3524:
3504:
3502:
3499:
3498:
3474:
3470:
3468:
3465:
3464:
3439:
3435:
3433:
3430:
3429:
3423:
3420:
3378:
3374:
3365:
3361:
3346:
3342:
3334:
3331:
3330:
3318:
3314:
3309:
3301:
3295:
3290:
3286:
3279:
3274:
3236:
3231:
3213:
3199:
3196: for
3194:
3188:
3184:
3178:
3174:
3165:
3161:
3155:
3151:
3136:
3132:
3126:
3122:
3110:
3106:
3097:
3093:
3081:
3077:
3071:
3067:
3062:
3050:
3046:
3031:
3027:
3026:
3022:
3013:
3009:
3007:
3004:
3003:
2992:
2978:
2970:
2966:
2961:
2959:
2952:symmetric group
2947:
2940:
2935:
2923:
2913:
2910:
2905:
2877:
2874:
2873:
2870:
2864:
2860:
2854:
2846:
2843:
2839:
2835:
2831:
2827:
2823:
2819:
2813:
2797:
2788:
2784:
2782:
2779:
2778:
2751:
2747:
2745:
2742:
2741:
2719:
2716:
2715:
2691:
2687:
2685:
2682:
2681:
2653:
2649:
2647:
2644:
2643:
2622:
2618:
2616:
2613:
2612:
2592:
2588:
2586:
2583:
2582:
2557:
2554:
2553:
2547:
2524:
2520:
2518:
2515:
2514:
2486:
2483:
2482:
2461:
2457:
2455:
2452:
2451:
2445:
2413:
2409:
2407:
2404:
2403:
2387:
2385:
2382:
2381:
2361:
2357:
2355:
2352:
2351:
2330:
2326:
2324:
2321:
2320:
2316:
2304:braid relations
2277:
2263:
2261:
2258:
2257:
2223:
2220:
2219:
2191:
2187:
2181:
2177:
2168:
2164:
2158:
2154:
2139:
2135:
2129:
2125:
2113:
2109:
2100:
2096:
2084:
2080:
2074:
2070:
2055:
2051:
2036:
2032:
2031:
2027:
2018:
2014:
2012:
2009:
2008:
1984:
1980:
1978:
1975:
1974:
1958:
1955:
1954:
1934:
1930:
1928:
1925:
1924:
1907:
1903:
1901:
1898:
1897:
1880:
1876:
1874:
1871:
1870:
1848:
1844:
1838:
1834:
1828:
1824:
1815:
1811:
1805:
1801:
1795:
1791:
1789:
1786:
1785:
1765:
1761:
1755:
1751:
1745:
1741:
1732:
1728:
1722:
1718:
1712:
1708:
1706:
1703:
1702:
1673:
1669:
1663:
1659:
1650:
1646:
1640:
1636:
1634:
1631:
1630:
1605:
1602:
1601:
1579:
1576:
1575:
1559:
1556:
1555:
1535:
1530:
1524:
1521:
1520:
1503:
1499:
1497:
1494:
1493:
1471:
1468:
1467:
1451:
1448:
1447:
1430:
1426:
1424:
1421:
1420:
1399:
1395:
1393:
1390:
1389:
1388:Every braid in
1382:
1366:
1362:
1360:
1357:
1356:
1352:
1336:
1332:
1330:
1327:
1326:
1322:
1306:
1302:
1300:
1297:
1296:
1258:
1253:
1199:
1191:Seifert circles
1187:
1147:J. W. Alexander
1127:
1121:
1100:
1097:
1096:
1093:homotopy groups
1076:
1073:
1072:
1052:
1049:
1048:
1030:
1027:
1026:
1010:
1007:
1006:
985:
982:
981:
965:
962:
961:
945:
942:
941:
907:
904:
903:
886:
882:
873:
869:
867:
864:
863:
846:
842:
840:
837:
836:
816:
813:
812:
796:
793:
792:
776:
773:
772:
756:
753:
752:
736:
733:
732:
716:
713:
712:
696:
693:
692:
669:
666:
665:
649:
646:
645:
642:symmetric group
625:
622:
621:
601:
598:
597:
580:
576:
574:
571:
570:
554:
551:
550:
534:
531:
530:
508:
505:
504:
479:
473:
453:quantum physics
433:fluid mechanics
429:
414:operation. The
394:
390:
388:
385:
384:
377:
373:
369:
179:
172:
169:
125:group operation
121:ambient isotopy
114:
92:
88:
86:
83:
82:
74:
73:braid group on
47:
43:
41:
38:
37:
28:
23:
22:
15:
12:
11:
5:
8845:
8835:
8834:
8829:
8824:
8819:
8802:
8801:
8799:
8798:
8786:
8773:
8770:
8769:
8767:
8766:
8764:Surgery theory
8761:
8756:
8751:
8746:
8741:
8736:
8731:
8726:
8721:
8716:
8711:
8706:
8701:
8696:
8691:
8686:
8680:
8678:
8674:
8673:
8671:
8670:
8665:
8663:Skein relation
8660:
8655:
8650:
8645:
8640:
8635:
8629:
8627:
8618:
8617:
8615:
8614:
8608:Unknotting no.
8605:
8600:
8595:
8594:
8593:
8583:
8578:
8577:
8576:
8571:
8566:
8561:
8556:
8546:
8541:
8536:
8531:
8526:
8521:
8516:
8511:
8506:
8501:
8500:
8499:
8489:
8484:
8483:
8482:
8472:
8467:
8461:
8459:
8453:
8452:
8450:
8449:
8443:
8434:
8428:
8419:
8413:
8404:
8400:
8394:
8390:
8384:
8380:
8374:
8370:
8363:
8361:
8355:
8354:
8352:
8351:
8346:
8345:
8344:
8339:
8328:
8326:
8320:
8319:
8317:
8316:
8311:
8305:
8296:
8290:
8281:
8275:
8269:
8263:
8259:
8253:
8249:
8243:
8239:
8233:
8229:
8225:
8221:
8217:
8213:
8207:
8203:
8197:
8193:
8186:
8184:
8178:
8177:
8161:
8160:
8153:
8146:
8138:
8129:
8128:
8126:
8125:
8120:
8115:
8110:
8105:
8100:
8095:
8089:
8087:
8083:
8082:
8080:
8079:
8074:
8069:
8063:
8061:
8057:
8056:
8049:
8048:
8041:
8034:
8026:
8020:
8019:
8011:Dance Your PhD
8000:
7989:
7974:
7964:on 4 June 2013
7950:
7935:
7922:
7906:
7905:External links
7903:
7901:
7900:
7895:
7875:
7870:
7851:
7818:
7777:(4): 273–302,
7761:
7748:"Braid theory"
7743:
7708:(1): 159–192,
7691:
7661:
7659:
7656:
7654:
7653:
7616:
7589:(2): 143–151.
7570:
7563:
7539:Ghrist, Robert
7530:
7528:
7527:
7481:
7446:(8): 979–991,
7428:
7399:hep-th/0201240
7392:(2): 558–563,
7347:(3): 529–553,
7326:Wilczek, Frank
7316:
7295:
7251:
7232:(1): 101–126.
7208:
7201:
7168:
7142:
7128:
7101:
7043:
7026:Markov, Andrey
7017:
6979:(11): 114101,
6963:
6895:
6853:(1): 277–304,
6830:
6801:(2): 285–315.
6785:
6765:
6763:
6760:
6759:
6758:
6753:
6748:
6743:
6738:
6733:
6728:
6721:
6718:
6717:
6716:
6705:
6702:
6697:
6694:
6691:
6687:
6681:
6678:
6675:
6671:
6667:
6662:
6659:
6656:
6652:
6646:
6643:
6640:
6636:
6632:
6627:
6624:
6621:
6617:
6611:
6608:
6605:
6601:
6577:
6574:
6571:
6568:
6565:
6562:
6559:
6552:
6549:
6545:
6518:
6514:
6487:
6482:
6462:
6450:
6445:
6440:
6435:
6432:
6427:
6423:
6400:
6396:
6371:
6367:
6363:
6358:
6346:
6345:
6334:
6331:
6328:
6323:
6318:
6313:
6310:
6305:
6301:
6297:
6292:
6288:
6284:
6281:
6278:
6275:
6272:
6267:
6263:
6259:
6256:
6253:
6248:
6244:
6240:
6237:
6232:
6228:
6224:
6219:
6215:
6200:
6199:
6187:
6184:
6181:
6178:
6168:
6164:
6160:
6155:
6151:
6147:
6142:
6137:
6132:
6127:
6123:
6119:
6116:
6111:
6107:
6103:
6100:
6097:
6094:
6091:
6086:
6082:
6078:
6075:
6072:
6069:
6064:
6059:
6054:
6051:
6046:
6042:
6018:
5996:
5991:
5973:
5959:
5955:
5934:
5910:
5907:
5904:
5901:
5898:
5895:
5868:
5847:
5844:
5843:
5842:
5831:
5828:
5825:
5822:
5819:
5809:
5805:
5801:
5796:
5792:
5788:
5782:
5778:
5775:
5771:
5765:
5761:
5757:
5754:
5742:minus the set
5721:
5717:
5692:
5689:
5686:
5683:
5679:
5675:
5672:
5669:
5666:
5657:to the points
5646:
5643:
5640:
5637:
5633:
5629:
5626:
5623:
5620:
5600:
5597:
5593:
5589:
5586:
5583:
5580:
5536:
5533:
5530:
5526:
5516:generators of
5505:
5502:
5499:
5477:
5473:
5463:generators of
5452:
5449:
5446:
5424:
5421:
5418:
5414:
5410:
5405:
5401:
5397:
5394:
5378:
5375:
5352:
5348:
5315:
5311:
5307:
5304:
5301:
5298:
5295:
5292:
5275:. Around 2001
5249: ≥ 5
5230:representation
5223:
5216:
5213:
5184:
5167:
5153:
5142:
5132:
5126:
5125:
5114:
5110:
5104:
5100:
5096:
5093:
5090:
5085:
5082:
5079:
5075:
5071:
5066:
5063:
5060:
5056:
5050:
5046:
5040:
5037:
5032:
5029:
5026:
5022:
5018:
5013:
5010:
5007:
5003:
4999:
4994:
4991:
4988:
4984:
4980:
4977:
4974:
4969:
4965:
4960:
4956:
4952:
4946:
4942:
4938:
4935:
4932:
4927:
4924:
4921:
4917:
4913:
4908:
4904:
4900:
4895:
4892:
4889:
4885:
4881:
4878:
4875:
4870:
4866:
4861:
4855:
4851:
4824:
4784:tensor product
4767:
4764:
4742:
4727:
4723:
4715:
4700:
4697:
4689:
4658:
4655:
4637:punctured disk
4626:
4619:
4616:
4589:
4584:The center of
4572:
4541:
4540:
4529:
4524:
4518:
4515:
4513:
4510:
4507:
4506:
4503:
4500:
4498:
4495:
4494:
4492:
4487:
4484:
4480:
4475:
4469:
4466:
4464:
4461:
4458:
4457:
4454:
4451:
4449:
4446:
4445:
4443:
4438:
4435:
4421:
4420:
4409:
4406:
4403:
4398:
4394:
4390:
4385:
4381:
4375:
4370:
4367:
4364:
4361:
4324:
4323:
4310:
4304:
4301:
4299:
4296:
4293:
4292:
4289:
4286:
4284:
4281:
4280:
4278:
4273:
4268:
4265:
4261:
4257:
4254:
4249:
4245:
4238:
4232:
4229:
4227:
4224:
4223:
4220:
4217:
4215:
4212:
4211:
4209:
4204:
4201:
4198:
4195:
4190:
4186:
4166:
4156:
4137:
4124:
4118:quotient group
4107:
4072:
4068:
4043:
4021:
4017:
3996:
3987:implying that
3985:
3984:
3973:
3970:
3965:
3962:
3957:
3953:
3949:
3944:
3940:
3936:
3931:
3928:
3923:
3919:
3915:
3910:
3906:
3882:
3860:
3856:
3852:
3847:
3843:
3831:
3830:
3816:
3812:
3806:
3802:
3798:
3795:
3791:
3786:
3782:
3776:
3772:
3766:
3762:
3758:
3755:
3723:
3719:
3694:
3691:
3686:
3682:
3678:
3675:
3649:
3645:
3633:quotient group
3629:
3628:
3616:
3612:
3608:
3605:
3602:
3598:
3595:
3592:
3588:
3583:
3579:
3575:
3571:
3568:
3565:
3561:
3558:
3531:
3527:
3523:
3520:
3517:
3513:
3510:
3507:
3477:
3473:
3442:
3438:
3422:
3418:
3415:
3407:
3406:
3395:
3392:
3387:
3384:
3381:
3377:
3373:
3368:
3364:
3360:
3355:
3352:
3349:
3345:
3341:
3338:
3312:
3293:
3284:
3277:
3267:
3266:
3255:
3251:
3247:
3244:
3239:
3234:
3230:
3226:
3223:
3220:
3216:
3212:
3209:
3206:
3202:
3191:
3187:
3181:
3177:
3173:
3168:
3164:
3158:
3154:
3150:
3145:
3142:
3139:
3135:
3129:
3125:
3119:
3116:
3113:
3109:
3105:
3100:
3096:
3090:
3087:
3084:
3080:
3074:
3070:
3065:
3059:
3056:
3053:
3049:
3045:
3042:
3039:
3034:
3030:
3025:
3021:
3016:
3012:
2990:
2974:
2964:
2955:
2945:
2938:
2909:
2906:
2904:
2901:
2900:
2899:
2887:
2884:
2881:
2866:
2856:
2851:abelianization
2841:
2837:
2833:
2829:
2825:
2815:
2800:
2796:
2791:
2787:
2772:
2754:
2750:
2729:
2726:
2723:
2712:
2709:Dehornoy order
2694:
2690:
2674:
2656:
2652:
2638:have infinite
2625:
2621:
2609:
2595:
2591:
2567:
2564:
2561:
2533:
2530:
2527:
2523:
2502:
2499:
2496:
2493:
2490:
2464:
2460:
2442:
2416:
2412:
2390:
2364:
2360:
2333:
2329:
2315:
2312:
2287:
2284:
2280:
2276:
2273:
2270:
2266:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2216:
2215:
2204:
2200:
2194:
2190:
2184:
2180:
2176:
2171:
2167:
2161:
2157:
2153:
2148:
2145:
2142:
2138:
2132:
2128:
2122:
2119:
2116:
2112:
2108:
2103:
2099:
2093:
2090:
2087:
2083:
2077:
2073:
2069:
2064:
2061:
2058:
2054:
2050:
2047:
2044:
2039:
2035:
2030:
2026:
2021:
2017:
1987:
1983:
1962:
1937:
1933:
1910:
1906:
1883:
1879:
1867:
1866:
1865:
1864:
1851:
1847:
1841:
1837:
1831:
1827:
1823:
1818:
1814:
1808:
1804:
1798:
1794:
1782:
1768:
1764:
1758:
1754:
1748:
1744:
1740:
1735:
1731:
1725:
1721:
1715:
1711:
1693:
1692:
1691:
1690:
1676:
1672:
1666:
1662:
1658:
1653:
1649:
1643:
1639:
1609:
1589:
1586:
1583:
1563:
1541:
1538:
1533:
1529:
1506:
1502:
1481:
1478:
1475:
1455:
1433:
1429:
1402:
1398:
1384:
1383:
1369:
1365:
1355:
1353:
1339:
1335:
1325:
1323:
1309:
1305:
1295:
1292:
1291:
1282:
1273:
1257:
1254:
1252:
1249:
1207:Wilhelm Magnus
1198:
1195:
1186:
1183:
1179:Markov theorem
1125:Brunnian braid
1120:
1117:
1104:
1080:
1056:
1034:
1014:
989:
969:
949:
929:
926:
923:
920:
917:
914:
911:
889:
885:
881:
876:
872:
849:
845:
820:
800:
780:
760:
740:
720:
700:
691:A path in the
673:
653:
629:
605:
583:
579:
558:
538:
512:
475:Main article:
472:
469:
437:chaotic mixing
428:
425:
397:
393:
376:is written as
364:
363:
356:
353:
346:
343:
333:
332:
324:
323:
316:
313:
306:
303:
293:
292:
280:
279:
276:
266:
265:
253:
252:
245:
242:
232:
231:
219:
218:
211:
208:
198:
197:
168:
165:
159:invariants of
95:
91:
50:
46:
26:
9:
6:
4:
3:
2:
8844:
8833:
8830:
8828:
8825:
8823:
8820:
8818:
8815:
8814:
8812:
8797:
8796:
8787:
8785:
8784:
8775:
8774:
8771:
8765:
8762:
8760:
8757:
8755:
8752:
8750:
8747:
8745:
8742:
8740:
8737:
8735:
8732:
8730:
8727:
8725:
8722:
8720:
8717:
8715:
8712:
8710:
8707:
8705:
8702:
8700:
8699:Conway sphere
8697:
8695:
8692:
8690:
8687:
8685:
8682:
8681:
8679:
8675:
8669:
8666:
8664:
8661:
8659:
8656:
8654:
8651:
8649:
8646:
8644:
8641:
8639:
8636:
8634:
8631:
8630:
8628:
8626:
8619:
8613:
8609:
8606:
8604:
8601:
8599:
8596:
8592:
8589:
8588:
8587:
8584:
8582:
8579:
8575:
8572:
8570:
8567:
8565:
8562:
8560:
8557:
8555:
8552:
8551:
8550:
8547:
8545:
8542:
8540:
8537:
8535:
8532:
8530:
8527:
8525:
8522:
8520:
8517:
8515:
8512:
8510:
8507:
8505:
8502:
8498:
8495:
8494:
8493:
8490:
8488:
8485:
8481:
8478:
8477:
8476:
8473:
8471:
8470:Arf invariant
8468:
8466:
8463:
8462:
8460:
8458:
8454:
8438:
8435:
8423:
8420:
8408:
8405:
8398:
8395:
8388:
8385:
8378:
8375:
8368:
8365:
8364:
8362:
8360:
8356:
8350:
8347:
8343:
8340:
8338:
8335:
8334:
8333:
8330:
8329:
8327:
8325:
8321:
8315:
8312:
8300:
8297:
8285:
8282:
8279:
8276:
8273:
8270:
8267:
8264:
8257:
8254:
8247:
8244:
8237:
8234:
8232:
8226:
8224:
8218:
8211:
8208:
8201:
8198:
8191:
8188:
8187:
8185:
8183:
8179:
8174:
8170:
8166:
8159:
8154:
8152:
8147:
8145:
8140:
8139:
8136:
8124:
8121:
8119:
8116:
8114:
8113:3D composites
8111:
8109:
8106:
8104:
8101:
8099:
8096:
8094:
8091:
8090:
8088:
8084:
8078:
8077:Brunnian link
8075:
8073:
8070:
8068:
8065:
8064:
8062:
8058:
8054:
8047:
8042:
8040:
8035:
8033:
8028:
8027:
8024:
8018:
8012:
8008:
8007:
8001:
7997:
7996:
7990:
7986:
7982:
7981:
7975:
7963:
7959:
7955:
7951:
7947:
7943:
7942:
7938:Macauley, M.
7936:
7934:
7930:
7926:
7923:
7919:
7918:
7913:
7912:"Braid group"
7909:
7908:
7898:
7892:
7888:
7884:
7880:
7876:
7873:
7867:
7863:
7862:
7857:
7852:
7849:
7845:
7840:
7835:
7831:
7827:
7823:
7819:
7816:
7812:
7808:
7804:
7800:
7796:
7792:
7788:
7784:
7780:
7776:
7772:
7771:
7766:
7762:
7759:
7755:
7754:
7749:
7744:
7741:
7737:
7733:
7729:
7725:
7721:
7716:
7711:
7707:
7703:
7702:
7697:
7692:
7690:
7684:
7679:
7675:
7671:
7667:
7663:
7662:
7649:
7645:
7641:
7637:
7630:
7626:
7620:
7612:
7608:
7604:
7600:
7596:
7592:
7588:
7584:
7580:
7574:
7566:
7564:9789814291408
7560:
7556:
7552:
7548:
7544:
7540:
7534:
7525:
7521:
7517:
7513:
7509:
7505:
7500:
7495:
7491:
7487:
7482:
7479:
7475:
7471:
7467:
7463:
7459:
7454:
7449:
7445:
7441:
7436:
7435:
7432:
7425:
7421:
7417:
7413:
7409:
7405:
7400:
7395:
7391:
7387:
7380:
7376:
7372:
7368:
7364:
7360:
7355:
7350:
7346:
7342:
7333:
7327:
7320:
7311:
7306:
7299:
7291:
7287:
7282:
7277:
7273:
7269:
7265:
7261:
7255:
7247:
7243:
7239:
7235:
7231:
7227:
7226:
7221:
7215:
7213:
7204:
7198:
7194:
7190:
7186:
7182:
7178:
7172:
7156:
7155:"Braid Index"
7152:
7146:
7139:
7135:
7131:
7125:
7121:
7117:
7116:
7111:
7105:
7098:
7094:
7090:
7086:
7082:
7078:
7073:
7068:
7064:
7060:
7059:
7054:
7047:
7039:
7035:
7031:
7027:
7021:
7014:
7010:
7005:
7000:
6995:
6990:
6986:
6982:
6978:
6974:
6967:
6960:
6956:
6952:
6948:
6944:
6940:
6936:
6932:
6928:
6924:
6919:
6914:
6911:(3): 036311,
6910:
6906:
6899:
6888:
6884:
6880:
6876:
6872:
6868:
6864:
6860:
6856:
6852:
6848:
6841:
6834:
6826:
6822:
6818:
6814:
6809:
6804:
6800:
6796:
6789:
6781:
6777:
6776:"Braid Group"
6770:
6766:
6757:
6754:
6752:
6749:
6747:
6744:
6742:
6739:
6737:
6734:
6732:
6729:
6727:
6724:
6723:
6703:
6700:
6695:
6692:
6689:
6685:
6679:
6676:
6673:
6669:
6665:
6660:
6657:
6654:
6650:
6644:
6641:
6638:
6634:
6630:
6625:
6622:
6619:
6615:
6609:
6606:
6603:
6599:
6591:
6590:
6589:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6550:
6547:
6543:
6534:
6516:
6512:
6503:
6485:
6470:
6467:
6443:
6430:
6425:
6421:
6398:
6394:
6384:
6365:
6361:
6332:
6321:
6308:
6303:
6299:
6290:
6286:
6282:
6273:
6270:
6265:
6261:
6254:
6246:
6242:
6238:
6230:
6226:
6217:
6213:
6205:
6204:
6203:
6182:
6179:
6176:
6166:
6162:
6158:
6153:
6149:
6145:
6140:
6130:
6125:
6121:
6117:
6109:
6105:
6101:
6098:
6095:
6092:
6089:
6084:
6080:
6070:
6062:
6049:
6044:
6040:
6032:
6031:
6030:
6016:
5994:
5979:
5957:
5953:
5932:
5924:
5921:, which is a
5905:
5902:
5899:
5893:
5885:
5882:
5866:
5859:
5853:
5829:
5823:
5820:
5817:
5807:
5803:
5799:
5794:
5790:
5786:
5776:
5773:
5763:
5759:
5745:
5744:
5743:
5741:
5737:
5719:
5715:
5706:
5705:inverse limit
5687:
5684:
5681:
5677:
5673:
5670:
5667:
5641:
5638:
5635:
5631:
5627:
5624:
5621:
5595:
5591:
5587:
5584:
5581:
5569:
5567:
5563:
5559:
5555:
5550:
5534:
5531:
5528:
5524:
5503:
5500:
5497:
5490:to the first
5475:
5471:
5450:
5447:
5444:
5422:
5419:
5416:
5412:
5403:
5399:
5395:
5392:
5384:
5374:
5372:
5368:
5350:
5346:
5336:
5330:
5313:
5309:
5302:
5299:
5296:
5290:
5283:of dimension
5282:
5278:
5274:
5266:
5265:Frank Wilczek
5262:
5258:
5257:Ruth Lawrence
5254:
5248:
5243:
5239:
5235:
5231:
5226:
5212:
5210:
5206:
5202:
5198:
5193:
5187:
5178:
5170:
5166:
5161:
5156:
5145:
5141:
5135:
5112:
5108:
5102:
5098:
5094:
5091:
5088:
5083:
5080:
5077:
5073:
5069:
5064:
5061:
5058:
5054:
5048:
5044:
5038:
5035:
5030:
5027:
5024:
5020:
5016:
5011:
5008:
5005:
5001:
4997:
4992:
4989:
4986:
4982:
4978:
4975:
4972:
4967:
4963:
4958:
4954:
4950:
4944:
4940:
4936:
4933:
4930:
4925:
4922:
4919:
4915:
4911:
4906:
4902:
4898:
4893:
4890:
4887:
4883:
4879:
4876:
4873:
4868:
4864:
4859:
4853:
4849:
4841:
4840:
4839:
4836:
4831:
4827:
4818:
4813:
4808:
4802:
4796:
4790:
4785:
4780:
4774:
4763:
4761:
4756:
4754:
4750:
4745:
4737:
4730:
4718:
4710:
4706:
4696:
4692:
4685:
4681:
4677:
4673:
4669:
4665:
4654:
4652:
4647:
4643:
4638:
4634:
4629:
4615:
4612:
4608:
4603:
4597:
4588:
4582:
4578:
4571:
4565:
4559:
4553:
4547:
4527:
4522:
4516:
4511:
4508:
4501:
4496:
4490:
4485:
4482:
4478:
4473:
4467:
4462:
4459:
4452:
4447:
4441:
4436:
4433:
4426:
4425:
4424:
4404:
4401:
4396:
4392:
4388:
4383:
4379:
4368:
4365:
4362:
4352:
4351:
4350:
4348:
4343:
4341:
4336:
4330:
4308:
4302:
4297:
4294:
4287:
4282:
4276:
4271:
4266:
4263:
4259:
4252:
4247:
4243:
4236:
4230:
4225:
4218:
4213:
4207:
4202:
4199:
4193:
4188:
4184:
4176:
4175:
4174:
4171:
4161:
4153:
4147:
4143:
4136:
4130:
4123:
4119:
4115:
4106:
4102:
4099: ⊂
4098:
4092:
4087:
4070:
4066:
4057:
4041:
4019:
4015:
3994:
3971:
3968:
3963:
3960:
3955:
3951:
3947:
3942:
3938:
3934:
3929:
3926:
3921:
3917:
3913:
3908:
3904:
3896:
3895:
3894:
3880:
3858:
3854:
3850:
3845:
3841:
3814:
3810:
3804:
3800:
3796:
3793:
3789:
3784:
3780:
3774:
3770:
3764:
3760:
3756:
3753:
3746:
3745:
3744:
3742:
3737:
3721:
3717:
3708:
3692:
3684:
3680:
3673:
3665:
3647:
3643:
3634:
3606:
3603:
3569:
3566:
3545:
3544:
3543:
3521:
3518:
3497:
3496:modular group
3493:
3475:
3471:
3458:
3440:
3436:
3427:
3414:
3412:
3393:
3385:
3382:
3379:
3375:
3366:
3362:
3353:
3350:
3347:
3343:
3336:
3329:
3328:
3327:
3326:
3321:
3315:
3307:
3304:
3296:
3287:
3280:
3272:
3253:
3249:
3245:
3242:
3237:
3232:
3228:
3224:
3221:
3218:
3210:
3207:
3204:
3189:
3185:
3179:
3175:
3171:
3166:
3162:
3156:
3152:
3148:
3143:
3140:
3137:
3133:
3127:
3123:
3117:
3114:
3111:
3107:
3103:
3098:
3094:
3088:
3085:
3082:
3078:
3072:
3068:
3057:
3054:
3051:
3047:
3043:
3040:
3037:
3032:
3028:
3023:
3019:
3014:
3010:
3002:
3001:
3000:
2998:
2993:
2986:
2982:
2977:
2973:
2967:
2958:
2953:
2948:
2941:
2934:
2931:
2926:
2921:
2916:
2885:
2882:
2879:
2869:
2859:
2852:
2845:is mapped to
2818:
2789:
2785:
2777:
2773:
2770:
2752:
2748:
2727:
2724:
2721:
2713:
2710:
2692:
2688:
2679:
2675:
2672:
2654:
2650:
2641:
2623:
2619:
2610:
2589:
2581:
2565:
2562:
2559:
2550:
2531:
2528:
2525:
2521:
2497:
2494:
2491:
2480:
2462:
2458:
2448:
2443:
2440:
2436:
2432:
2414:
2410:
2380:
2362:
2358:
2349:
2331:
2327:
2318:
2317:
2311:
2309:
2305:
2301:
2285:
2282:
2274:
2271:
2268:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2202:
2198:
2192:
2188:
2182:
2178:
2174:
2169:
2165:
2159:
2155:
2151:
2146:
2143:
2140:
2136:
2130:
2126:
2120:
2117:
2114:
2110:
2106:
2101:
2097:
2091:
2088:
2085:
2081:
2075:
2071:
2067:
2062:
2059:
2056:
2052:
2048:
2045:
2042:
2037:
2033:
2028:
2024:
2019:
2015:
2007:
2006:
2005:
2003:
1985:
1981:
1960:
1951:
1935:
1931:
1908:
1904:
1881:
1877:
1849:
1845:
1839:
1835:
1829:
1825:
1821:
1816:
1812:
1806:
1802:
1796:
1792:
1783:
1766:
1762:
1756:
1752:
1746:
1742:
1738:
1733:
1729:
1723:
1719:
1713:
1709:
1700:
1699:
1698:
1697:
1696:
1674:
1670:
1664:
1660:
1656:
1651:
1647:
1641:
1637:
1628:
1627:
1626:
1625:
1624:
1621:
1607:
1587:
1584:
1581:
1561:
1539:
1536:
1531:
1527:
1504:
1500:
1479:
1476:
1473:
1453:
1431:
1427:
1418:
1400:
1396:
1367:
1363:
1354:
1337:
1333:
1324:
1307:
1303:
1294:
1293:
1288:
1283:
1279:
1274:
1270:
1265:
1264:
1261:
1248:
1246:
1242:
1237:
1235:
1231:
1227:
1223:
1222:presentations
1218:
1216:
1212:
1211:Adolf Hurwitz
1208:
1204:
1194:
1192:
1182:
1180:
1175:
1173:
1169:
1168:Vaughan Jones
1165:
1163:
1159:
1154:
1152:
1148:
1144:
1140:
1136:
1132:
1126:
1119:Closed braids
1116:
1115:are trivial.
1102:
1094:
1078:
1070:
1054:
1046:
1032:
1012:
1001:
987:
967:
947:
927:
924:
921:
918:
915:
912:
909:
887:
883:
879:
874:
870:
847:
843:
834:
818:
798:
778:
758:
738:
718:
698:
689:
687:
671:
651:
643:
627:
619:
603:
581:
577:
556:
536:
528:
527:
510:
503:
498:
496:
492:
488:
484:
478:
468:
466:
462:
458:
454:
449:
447:
443:
438:
434:
424:
421:
417:
413:
395:
391:
381:
361:
357:
354:
351:
347:
344:
341:
337:
336:
331:
330:
329:
321:
317:
314:
311:
307:
304:
301:
297:
296:
291:
290:
289:
287:
277:
274:
270:
269:
264:
263:
262:
260:
250:
246:
243:
240:
236:
235:
230:
229:
228:
226:
216:
212:
209:
206:
202:
201:
196:
195:
194:
192:
188:
182:
175:
164:
162:
158:
154:
150:
146:
142:
138:
134:
130:
126:
123:), and whose
122:
118:
111:
93:
89:
80:
77:
70:
48:
44:
34:
30:
19:
8817:Braid groups
8793:
8781:
8709:Double torus
8694:Braid theory
8509:Crossing no.
8504:Crosscap no.
8190:Figure-eight
8103:Braided rope
8072:Braid theory
8066:
8010:
8005:
7994:
7985:the original
7979:
7966:. Retrieved
7962:the original
7945:
7940:
7915:
7889:, Elsevier,
7886:
7864:, Springer,
7861:Braid Groups
7860:
7829:
7825:
7774:
7768:
7751:
7705:
7699:
7673:
7666:Birman, Joan
7639:
7636:Mat. Zametki
7635:
7619:
7586:
7582:
7573:
7542:
7533:
7499:math/0303042
7492:(1): 21–29,
7489:
7485:
7453:math/0201303
7443:
7439:
7431:
7389:
7385:
7344:
7340:
7331:
7319:
7298:
7271:
7267:
7254:
7229:
7223:
7184:
7171:
7159:. Retrieved
7145:
7114:
7104:
7072:math/0405498
7062:
7056:
7046:
7037:
7033:
7020:
6976:
6972:
6966:
6918:nlin/0510075
6908:
6904:
6898:
6887:the original
6850:
6846:
6833:
6798:
6794:
6788:
6779:
6769:
6465:
6385:
6347:
6201:
5855:
5740:Hilbert cube
5570:
5551:
5383:direct limit
5380:
5334:
5328:
5261:Chetan Nayak
5246:
5221:
5218:
5191:
5182:
5176:
5168:
5164:
5151:
5143:
5139:
5130:
5127:
4834:
4822:
4816:
4806:
4800:
4794:
4788:
4778:
4772:
4769:
4760:cryptography
4757:
4748:
4740:
4728:
4713:
4705:word problem
4702:
4690:
4676:braid theory
4660:
4648:
4641:
4624:
4621:
4610:
4601:
4595:
4593:is equal to
4586:
4583:
4576:
4569:
4563:
4557:
4551:
4545:
4542:
4422:
4347:presentation
4344:
4334:
4328:
4325:
4169:
4159:
4145:
4141:
4134:
4128:
4121:
4104:
4100:
4096:
4090:
3986:
3832:
3738:
3630:
3462:
3408:
3319:
3310:
3308:and denoted
3302:
3299:
3291:
3282:
3275:
3268:
2988:
2984:
2980:
2975:
2971:
2962:
2956:
2943:
2936:
2924:
2914:
2911:
2903:Interactions
2867:
2857:
2816:
2776:homomorphism
2678:linear order
2671:torsion-free
2579:
2548:
2477:embeds as a
2446:
2435:trefoil knot
2379:cyclic group
2303:
2300:Artin groups
2217:
2002:presentation
1952:
1868:
1694:
1622:
1387:
1259:
1241:braid theory
1238:
1219:
1200:
1188:
1176:
1166:
1155:
1151:string links
1142:
1134:
1130:
1128:
1004:
1002:
832:
690:
524:
499:
480:
450:
430:
427:Applications
382:
367:
327:
285:
283:
258:
256:
224:
222:
190:
186:
180:
173:
170:
167:Introduction
119:(e.g. under
109:
75:
72:
66:
29:
18:Braid theory
8822:Knot theory
8544:Linking no.
8465:Alternating
8266:Conway knot
8246:Carrick mat
8200:Three-twist
8165:Knot theory
8067:Braid group
7832:: 119–126,
7642:: 227–231.
7310:0711.3941v2
7274:: 119–126.
7220:Artin, Emil
7004:10919/24513
6746:Knot theory
5734:and to the
5255:. In 1990,
4709:normal form
4132:. We claim
4054:denote the
3741:isomorphism
3662:modulo its
3298:called the
2920:permutation
2812:defined by
2774:There is a
2707:called the
1232:of certain
1217:from 1891.
1213:'s work on
1185:Braid index
1160:. In 1935,
831:-tuples of
485:concept of
133:knot theory
69:mathematics
8811:Categories
8704:Complement
8668:Tabulation
8625:operations
8549:Polynomial
8539:Link group
8534:Knot group
8497:Invertible
8475:Bridge no.
8457:Invariants
8387:Cinquefoil
8256:Perko pair
8182:Hyperbolic
8108:3D weaving
7968:1 November
7917:PlanetMath
7822:Fox, Ralph
7260:Fox, Ralph
6762:References
6500:. In 1968
5976:unordered
5923:CW complex
5850:See also:
5846:Cohomology
5558:completion
5554:topologies
4682:and every
4112:, it is a
2930:surjective
2769:free group
2431:knot group
1419:the group
1226:Emil Artin
1203:Emil Artin
1172:polynomial
1123:See also:
731:points of
549:copies of
155:); and in
8598:Stick no.
8554:Alexander
8492:Chirality
8437:Solomon's
8397:Septafoil
8324:Satellite
8284:Whitehead
8210:Stevedore
7815:123680847
7799:0020-9910
7758:EMS Press
7715:0711.3785
7611:123442457
7424:119388336
7328:(1996), "
6696:ℓ
6686:ω
6670:ω
6651:ω
6639:ℓ
6635:ω
6620:ℓ
6616:ω
6610:ℓ
6600:ω
6573:≤
6561:≤
6544:ω
6431:
6309:
6291:∗
6247:∗
6218:∗
6180:≠
6159:≠
6131:∈
6050:
5821:≠
5787:∣
5777:∈
5501:−
5448:−
5437:send the
5409:→
5396::
5369:over the
5300:−
5092:…
5036:−
4990:−
4976:…
4934:…
4891:−
4877:…
4850:σ
4575:→ PSL(2,
4509:−
4460:−
4408:⟩
4360:⟨
4295:−
4264:−
4256:↦
4244:σ
4197:↦
4185:σ
4144:≅ PSL(2,
4086:generated
3961:−
3952:σ
3939:σ
3927:−
3918:σ
3905:σ
3811:σ
3801:σ
3781:σ
3771:σ
3761:σ
3743:. Define
3587:→
3582:¯
3391:→
3383:−
3372:→
3359:→
3351:−
3340:→
3219:≥
3208:−
3055:−
3041:…
2795:→
2725:≥
2594:∞
2563:≥
2481:into the
2283:≥
2272:−
2241:−
2235:≤
2229:≤
2189:σ
2179:σ
2166:σ
2156:σ
2137:σ
2127:σ
2111:σ
2098:σ
2082:σ
2072:σ
2068:∣
2060:−
2053:σ
2046:…
2034:σ
1932:σ
1905:σ
1878:σ
1846:σ
1836:σ
1826:σ
1813:σ
1803:σ
1793:σ
1763:σ
1753:σ
1743:σ
1730:σ
1720:σ
1710:σ
1671:σ
1661:σ
1648:σ
1638:σ
1608:σ
1537:−
1528:σ
1501:σ
1364:σ
1334:σ
1304:σ
1245:Ralph Fox
1215:monodromy
925:≤
913:≤
191:different
157:monodromy
81:(denoted
8783:Category
8653:Mutation
8621:Notation
8574:Kauffman
8487:Brunnian
8480:2-bridge
8349:Knot sum
8280:(12n242)
8086:Practice
8053:Braiding
7858:(2008),
7740:16467487
7627:(1969).
7524:13892069
7478:16998867
7379:18726223
7179:(1974).
7161:6 August
7112:(1974),
7097:14494095
7028:(1935),
7013:21469863
6951:16605655
6883:47710742
6825:14502859
6720:See also
5242:faithful
4792:and let
4782:-folded
4726:, ..., σ
4543:Mapping
4094:, since
4056:subgroup
3250:⟩
3024:⟨
2863:, then
2642:; i.e.,
2479:subgroup
2199:⟩
2029:⟨
1417:generate
902:for all
833:distinct
502:manifold
483:homotopy
286:composed
225:the same
193:braids:
8795:Commons
8714:Fibered
8612:problem
8581:Pretzel
8559:Bracket
8377:Trefoil
8314:L10a140
8274:(11n42)
8268:(11n34)
8236:Endless
8123:Weaving
7848:0150755
7807:0422673
7779:Bibcode
7732:2747726
7648:0242196
7603:0274463
7516:2204494
7470:2196643
7404:Bibcode
7359:Bibcode
7290:0150755
7246:1969218
7138:0375281
7089:1465027
7040:: 73–78
6981:Bibcode
6959:7142834
6943:2231368
6923:Bibcode
6875:1742169
6855:Bibcode
6466:ordered
5972:is the
5738:of the
4820:. Then
4766:Actions
4173:map to
3494:of the
3490:is the
3455:is the
3306:strands
2578:is the
2433:of the
2348:trivial
1197:History
1045:strings
420:inverse
227:braid:
117:-braids
79:strands
8759:Writhe
8729:Ribbon
8564:HOMFLY
8407:Unlink
8367:Unknot
8342:Square
8337:Granny
8060:Theory
7893:
7868:
7846:
7813:
7805:
7797:
7738:
7730:
7646:
7609:
7601:
7561:
7543:Braids
7522:
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7095:
7087:
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6957:
6949:
6941:
6881:
6873:
6823:
6461:, the
5253:linear
4749:CHEVIE
4607:kernel
4423:where
4326:where
4152:cosets
4034:. Let
3664:center
3271:kernel
2987:+1) ∈
2402:, and
1784:(iib)
1701:(iia)
1135:closed
616:-fold
596:, the
457:anyons
143:where
139:); in
71:, the
8749:Twist
8734:Slice
8689:Berge
8677:Other
8648:Flype
8586:Prime
8569:Jones
8529:Genus
8359:Torus
8173:links
8169:knots
8093:Braid
7811:S2CID
7736:S2CID
7710:arXiv
7704:, 3,
7687:. In
7678:arXiv
7632:(PDF)
7607:S2CID
7520:S2CID
7494:arXiv
7474:S2CID
7448:arXiv
7420:S2CID
7394:arXiv
7375:S2CID
7349:arXiv
7305:arXiv
7242:JSTOR
7093:S2CID
7067:arXiv
6955:S2CID
6913:arXiv
6890:(PDF)
6879:S2CID
6843:(PDF)
6821:S2CID
6803:arXiv
6300:UConf
6041:UConf
5269:SO(3)
5199:is a
4639:with
4635:of a
2640:order
1129:When
1069:up to
1025:with
686:orbit
493:of a
412:group
187:braid
151:(see
145:Artin
8754:Wild
8719:Knot
8623:and
8610:and
8591:list
8422:Hopf
8171:and
7970:2007
7891:ISBN
7866:ISBN
7795:ISSN
7559:ISBN
7197:ISBN
7163:2014
7124:ISBN
7009:PMID
6947:PMID
6567:<
6422:Conf
5856:The
5566:disk
5332:and
5263:and
5137:and
4830:acts
4703:The
4684:link
4680:knot
4668:knot
4664:link
4555:and
4332:and
4163:and
3269:The
2714:For
2444:The
1923:and
1629:(i)
1466:and
1177:The
1158:knot
1139:link
919:<
372:and
8739:Sum
8260:161
8258:(10
7931:'s
7834:doi
7787:doi
7720:doi
7706:102
7591:doi
7551:doi
7504:doi
7458:doi
7412:doi
7367:doi
7345:479
7276:doi
7234:doi
7189:doi
7077:doi
6999:hdl
6989:doi
6977:106
6931:doi
6863:doi
6851:403
6813:doi
6471:of
6413:is
5980:of
5232:is
5189:on
4832:on
4814:of
4674:in
4561:to
4549:to
4088:by
4058:of
3709:of
3635:of
2979:= (
2922:on
2861:↦ k
2820:↦ 1
2680:on
2669:is
2346:is
1519:or
1095:of
644:on
620:of
529:of
259:not
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67:In
8813::
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380:.
378:στ
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4219:1
4214:1
4208:[
4203:=
4200:R
4194:C
4189:1
4170:C
4167:2
4165:σ
4160:C
4157:1
4155:σ
4148:)
4146:Z
4142:C
4140:/
4138:3
4135:B
4129:C
4127:/
4125:3
4122:B
4110:)
4108:3
4105:B
4103:(
4101:Z
4097:C
4091:c
4071:3
4067:B
4042:C
4020:3
4016:B
3995:c
3972:c
3969:=
3964:1
3956:2
3948:c
3943:2
3935:=
3930:1
3922:1
3914:c
3909:1
3881:c
3859:3
3855:b
3851:=
3846:2
3842:a
3829:.
3815:2
3805:1
3797:=
3794:b
3790:,
3785:1
3775:2
3765:1
3757:=
3754:a
3722:3
3718:B
3693:,
3690:)
3685:3
3681:B
3677:(
3674:Z
3648:3
3644:B
3627:.
3615:)
3611:R
3607:,
3604:2
3601:(
3597:L
3594:S
3591:P
3578:)
3574:R
3570:,
3567:2
3564:(
3560:L
3557:S
3530:)
3526:Z
3522:,
3519:2
3516:(
3512:L
3509:S
3506:P
3476:3
3472:B
3441:3
3437:B
3419:3
3386:1
3380:n
3376:P
3367:n
3363:P
3354:1
3348:n
3344:F
3337:1
3320:n
3313:n
3311:P
3303:n
3294:n
3292:B
3285:n
3283:S
3278:n
3276:B
3254:.
3246:1
3243:=
3238:2
3233:i
3229:s
3225:,
3222:2
3215:|
3211:j
3205:i
3201:|
3190:i
3186:s
3180:j
3176:s
3172:=
3167:j
3163:s
3157:i
3153:s
3149:,
3144:1
3141:+
3138:i
3134:s
3128:i
3124:s
3118:1
3115:+
3112:i
3108:s
3104:=
3099:i
3095:s
3089:1
3086:+
3083:i
3079:s
3073:i
3069:s
3064:|
3058:1
3052:n
3048:s
3044:,
3038:,
3033:1
3029:s
3020:=
3015:n
3011:S
2991:n
2989:S
2985:i
2981:i
2976:i
2972:s
2965:n
2963:B
2957:i
2946:n
2944:S
2939:n
2937:B
2925:n
2915:n
2886:0
2883:=
2880:k
2868:i
2865:σ
2858:i
2855:σ
2842:3
2840:σ
2838:2
2836:σ
2834:1
2832:σ
2830:3
2828:σ
2826:2
2824:σ
2817:i
2814:σ
2799:Z
2790:n
2786:B
2753:n
2749:B
2728:3
2722:n
2711:.
2693:n
2689:B
2673:.
2655:n
2651:B
2624:n
2620:B
2608:.
2590:B
2566:1
2560:n
2549:n
2532:1
2529:+
2526:n
2522:B
2501:)
2498:1
2495:+
2492:n
2489:(
2463:n
2459:B
2447:n
2441:.
2415:3
2411:B
2389:Z
2363:2
2359:B
2332:1
2328:B
2286:2
2279:|
2275:j
2269:i
2265:|
2244:2
2238:n
2232:i
2226:1
2203:,
2193:i
2183:j
2175:=
2170:j
2160:i
2152:,
2147:1
2144:+
2141:i
2131:i
2121:1
2118:+
2115:i
2107:=
2102:i
2092:1
2089:+
2086:i
2076:i
2063:1
2057:n
2049:,
2043:,
2038:1
2025:=
2020:n
2016:B
1986:n
1982:B
1961:n
1936:3
1909:2
1882:1
1850:3
1840:2
1830:3
1822:=
1817:2
1807:3
1797:2
1781:,
1767:2
1757:1
1747:2
1739:=
1734:1
1724:2
1714:1
1689:,
1675:1
1665:3
1657:=
1652:3
1642:1
1588:1
1585:+
1582:i
1562:i
1540:1
1532:i
1505:i
1480:1
1477:+
1474:i
1454:i
1432:4
1428:B
1401:4
1397:B
1368:3
1338:2
1308:1
1143:n
1131:X
1103:Y
1079:X
1055:Y
1033:n
1013:X
988:Y
968:n
948:Y
928:n
922:j
916:i
910:1
888:j
884:x
880:=
875:i
871:x
848:n
844:X
819:n
799:Y
779:n
759:n
739:X
719:n
699:n
672:n
652:n
628:X
604:n
582:n
578:X
557:X
537:n
511:X
396:4
392:B
374:τ
370:σ
181:n
174:n
115:n
94:n
90:B
76:n
63:.
49:5
45:B
20:)
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